let's see how quickly we can go through everything you need to know for OCR level physics we can represent forces with vectors that is an arrow that shows the direction and magnitude of the force if two forces act on an object there's a resultant force we find this by technically adding the vectors however if they go in opposite directions one must be negative so in this case the resultant force would be 3 newtons to the right and that's positive if we've decided that positive is in the right direction if vectors are at right angles to each other you use Pythagoras to find the resultant this works because you can make a triangle by moving one of the forces you could also be expected to use trig that's so to find either one of these angles if a measurement or quantity just has magnitude but no direction it's not a vector but it's called a scalar instead here are some examples of both note that displacement is distance traveled with a direction similarly velocity is the vector form of speed weight is another name for the force due to gravity that acts on an object it's calculated by multiplying the mass in kilograms by gravitational field strength or G which here on Earth is 9.8 newtons per kg now if you hold an object up with your hand you must be pushing up with a force that is equal to its weight in order for the forces to be balanced and so it doesn't accelerate however that means that if you lift it upwards at a constant speed that's also true we can therefore then calculate the energy that is used to lift this object using the equation for work done that's work done equals force time distance moved this equation is true for any situation but in this case the force is the weight and the distance is the height so we could say the gain in energy is equal to mass * g * h does that look familiar it should because that's the exact same equation for calculating gravitational potential energy that's GP gained to be precise to calculate work done the force and distance moved must be parallel if the force is at an angle however we must resolve it to find the component that is parallel to the distance that's why sometimes you'll see a cos theta on the end of the work done equation in your formula sheet if we divide both sides of the equation by time we end up with power equals force time velocity we sometimes call this power developed just tells you how much work is done per second by or against the force newton's first law is this when there's no resultant force an object's motion is constant in other words no change in velocity that could be because there's no forces acting or the forces are balanced newton's second law involves unbalanced forces that is there is a resultant force this is equal to ma mass times acceleration only one of these can be true in any situation there's either no resultant force or there is newton's third law however is always true and this is the one that people get confused about understandably for every action or force there is an equal and opposite reaction force but this is not referring to balanced forces it's all about perspective when we think about the first two laws we only really consider the object itself for example the force pulling downwards on the ball is its weight we know that the earth is pulling down the ball but Newton's third law says the complete opposite is true as well the ball is also pulling the earth up now the Earth is so massive that it doesn't really have an effect but it's still true nevertheless another example if you have two ice skaters if the guy skater pushes on the girl skater there's an equal and opposite reaction force pushing back on him too that's why they both move away from where they were if an object is on a slope its weight mg will try to pull it downwards however the reaction force of the slope pushes perpendicular to the slope this is equal to mg cos theta where theta is the angle of the ramp for an explanation of where the angles come from check out my mass on a slope video the combination of these two forces result in a force pulling down parallel to the slope this is equal to mg sin theta that's always going to be true no matter what the object is doing if it's stationary or traveling at a constant speed that must mean another force is pushing or pulling up the slope that has to be equal to this component mg sin theta quite often that's friction no matter what forces are involved you find the resultant like you do for any other situation the sum of all forces in one direction minus the sum of all forces in the opposite direction then you can use this resultant force in F= ma if you're given the height of the ramp chances are you're going to have to use energy too maybe the mass is sliding down the ramp for example if there's no frictional forces we can say mgh at the top equals half mv squared at the bottom as per usual if you're told the speed at the bottom chances are some energy is being lost instead so we just use the work done equation with the work done being that energy lost and that can give us the average frictional force just remember that the distance has to be the distance traveled not the height in that case if an object is accelerating we use one of Newton's equations of motion and their suvat variables if an object is only moving in one direction s u and v can all be positive even if it's downwards you do have to be careful with a though as that could be negative for example if the object is decelerating in any question you write down your variables and the unknown you need to find cross out the unused fifth variable choose the right equation and rearrange if necessary to deal with an object being thrown upwards and returning to its original position S is technically zero and things get a bit messy then in this situation it's always a good idea to split the problem in half from start to the apex where V ends up being zero and then just double the time if the question requires it for objects being fired horizontally off the edge of something say a cliff you split the problem into vertical and horizontal use suvat for vertical where u is zero as the initial speed is only horizontal a is equal to g 9.81 and s is the height it falls chances are you're going to use s= ut +/ a t ^2 u disappears because u is zero you don't use suvat for the horizontal motion though as its horizontal velocity stays at that initial speed so horizontally you just use speed equals distance over time for objects fired at an angle just use my easy vectors trick to resolve the initial speed into vertical and horizontal components link in description if you haven't watched that yet again use suvac for vertical speed equals distance over time for horizontal like the ball being thrown upwards and coming back down you might need to split the trajectory in half and just go from the start to the apex momentum is equal to mass times velocity the unit therefore is kilogram m/s momentum is a vector which means you have negative momentum if your velocity is negative in a collision kinetic energy isn't always conserved but total momentum always is calculations on this can be tricky but you just have to be careful with your pluses and minuses you write down m1 U1 if there's just one object moving to begin with remember U from SUVA we can use it here too and on m2 U2 if there's a second object moving too this then is the total momentum before the collision before the event this could also be zero if nothing's moving to begin with say a cannon about to fire then all we have to say is that this is equal to the total momentum afterwards m1 v1 for one object plus m2 v2 if there's a second object moving too if they've coupled together we just say m * v where m is the total mass of the two then all you have to do is pop your numbers in making sure that everything traveling to the left say has a negative velocity and you'll be left with one unknown rearrange to find it you get your answer incidentally in the case of the cannon as there's zero total momentum before the same must be true after two even though the cannonball is moving that must mean the cannon has the same momentum but in the opposite direction so they still add up to zero this is an example of recoil one last and very important example rebound when an object or particle collides with an object that essentially doesn't move say a wall it will rebound with the same speed but in the opposite direction that's not a zero change in momentum though as momentum is a vector if its initial momentum is mu where u is the initial velocity then the change in momentum also known as impulse will be -2 mu of course total energy is always conserved too but some of that will likely be lost to the surroundings as heat sound etc total kinetic energy is very rarely conserved as a result we call such collisions inelastic like two plasterine balls smooshing together there's no kinetic energy afterwards so that's as inelastic as you can get if the conditions and the objects are just right though you can get collisions where the total kinetic energy is conserved in theory anyway these are elastic collisions an example of this would be snooker balls hitting each other or a Newton's cradle these collisions are very close to being elastic so we can treat them as such total kinetic energy is conserved force and momentum are closely linked newton's second law says that F= ma but we also know that A is equal to delta V over T so actually it's also true that force is equal to change in momentum over time or we can say the rate of change of momentum that's why the alternative unit for momentum is Newton seconds impulse is the alternative name for change in momentum so it has the same unit if we draw a graph of how the force acting on an object changes over time the area under the graph gives us the impulse so if the object was stationary initially it will now have that momentum due to the impulse provided to it if a fluid like air or water pushes on something at a constant rate say a hose pipe spraying a wall or air through a wind turbine we can find the momentum carried per second by it in reality this is the same as force as you know from our equation units are your friends here as you won't be given an equation for this so let's take F= delta MV over T but instead of swapping v / t for acceleration we split it into m /t * v m /t is the mass in kilog moving per second you might be given that but you'll more likely be given the density of the fluid instead and the cross-sectional area that it's moving through so mass is equal to density time volume and volume is equal to area time length length over time gives you speed so the most useful equation for this is f= row av^ 2 a moment or torque is a turning force around a pivot it's calculated by force times distance to the pivot so its unit is Newton meters however it's not the same as work done as force and distance must be perpendicular instead of parallel if they're not perpendicular you must resolve one of them usually the force by multiplying by s or cos theta depending on what angle you're given again you can use my easy vectors trick in short we multiply by cos theta if the angle is between the resultant and the component sign if not the principle of moments is this for an object to be in equilibrium the sum of clockwise moments must equal the sum of anticlockwise moments bear in mind that an object can already be spinning and this still be the case in reality there is another condition needed for an object to be in equilibrium which you already know there also can be no resultant force acting on the object it turns out that the principle of moments can be applied to any scenario even if there appears to be no pivot as such in fact you can make any point your imaginary pivot and it still works for example this guy standing on a plank resting on two supports you first take moments about one of the supports to eliminate it from the equation taking moments about the left support the moments due to the weight of the plank this is easy to forget by the way and the dude going clockwise must be balanced by the moment due to the reaction force from the support on the right you can find the reaction force this way and then you can use this with just balanced forces to calculate the reaction force on the left you don't need to use moments if you just have one unknown force a couple is defined as two forces pushing in opposite directions on an object such that they produce a resultant moment but no resultant force meaning that the object will start to spin if it wasn't already but it will not accelerate it's not in equilibrium as it's only satisfied one of our two conditions an object will topple over if its center of gravity has moved past the pivot vertically so the moment due to its weight has changed direction from clockwise to anticlockwise or vice versa up thrust is the force that a fluid pushes upwards with this is due to the fact that the object displaces the same volume of liquid as its own volume that means that up thrust is equal to the weight of the liquid displaced and m is equal to row * v density of liquid time volume of the object of course the weight of the object also pulls downwards the resultant force is 1 minus the other which can then be used in Newton's second law as per usual if the object is moving however there will be drag pushing in the opposite direction to its velocity too which also must be taken into consideration when finding the resultant force stokes law gives us the magnitude of this drag force f= 6 pi eta rv sometimes you'll see a mu instead of eta by the way this only works when the object moving through the fluid is a sphere of radius r and it has a smooth surface and it's traveling at speed v through the fluid and we assume that it results in lamina flow that means that the fluid is disturbed as little as possible as it moves it's not turbulent flow eta or mu is the dynamic viscosity it's just a constant associated with the fluid density is equal to mass divided by volume so the unit we have is kg per me cubed you know that from GCSE just be careful when it comes to alloys and other mixtures all you have to remember is that the total mass is equal to the mass of material 1 plus the mass of material 2 etc then replace the masses with density time volume and it all should work out forces can also deform an object if you pull on a spring that is fixed at one end it will stretch or extend hook's law states that F= K that's force equals spring constant sometimes called stiffness times extension the unit for spring constant is Newtons per meter this works for any object that stretches elastically that is returns to its original shape once the force is removed it can also be true if an object is compressed instead the area under the graph gives you the work done on the spring and therefore the elastic potential energy stored in the spring if you attach two springs in series we might say end to end they stretch twice as far with the same force meaning the spring constant must be half what it is for one of the springs side by side in parallel and the opposite is true even if not a spring materials can still be stretched or compressed they can be deformed with a force stress is basically pressure it's force per unit area so newtons per meter squared also called Pascals strain is just a ratio of extension to original length drawing a graph of stress against strain will likely give you a proportional relationship at least to begin with the gradient of this graph gives us the Young modulus every material has its own unique number as strain doesn't have a unit the Young modulus is also measured in Pascals replacing stress and strain with their own equations the full version of the equation is E= L / A delt or X or E depending on what you use for extension area is proportional to D² as area is equal to P D^2 over 4 that means if you double the diameter the extension quarters add more force again and the graph starts to curve though we've gone past the limit of proportionality we can't use Young's modulus past this point however even after it curves if we remove the force it can still return to its original position it can still behave elastically we say the unloading curve will look different to the loading curve if we make a graph of force against extension the area under the loading curve is still the energy needed to stretch the material just like with springs the area under the unloading curve is the energy that you get out as it's returned to its original length that means that the area enclosed between the two is the energy lost usually as heat increase the force even more and it won't return to its original length we've gone past the elastic limit it's now undergoing plastic deformation instead of elastic the maximum stress the curve goes to is called the ultimate tensile stress the line can go down after this point as the wire starts to neck it carries on extending even if you remove some of the force a ductile material like copper will have a large plastic region whereas a brittle material generally has a high young modulus and suddenly snaps before being able to deform plastically for something moving in a circle only one condition needs to be satisfied there must be a force that continuously acts at 90° to the object's velocity or direction of motion that means that its velocity and direction of travel constantly changes but it still travels at a constant speed as velocity is a vector it's still accelerating because the direction of the velocity is changing even though the magnitude of the velocity the speed is not this centripedal acceleration is equal to v ^2 / r where r is the radius of the circle as f= ma the centripedal force is equal to mv^2 / r as speed is equal to distance over time we can calculate it by doing circumference divided by the time period so 2 pi r / t it's also useful to consider its angular velocity also known as angular speed or angular frequency that's how many radians of the orbit are covered every second we give it the symbol omega a curly w as that's equal to 2i / t how many radians/s that means that v = omega r this is a very important equation that people often forget as normal frequency is 1 over the time period omega also equals 2 pi f replacing v in our cental acceleration and force equation with this we end up with the alternative equations a= omega^ 2 r and f= m omega 2 r if you're given the time period you use the omega version and replace the omega with 2 pi over t if you're given the actual speed though you use the first equation a= v ^2 r or f= mv² over r similar to angular velocity angular displacement is just how many radians an object has turned through like we know from waves to turn degrees into radians we divide by 360 to turn into whole circles or whole cycles then multiply by 2 pi to turn it into radians if it's a vertical loop like a loop toloop on a roller coaster you have to take into account the support force or reaction force of the track we might call that S at the bottom mg is pulling down and S is pushing up the centrial force is one minus the other the resultant of the two forces at the top both are pulling down meaning the centrial force equals S plus mg instead in both cases we usually then equate these to mv^ 2 / r when it's at either side weight isn't contributing to the centrial force at all therefore S is equal to MV^2 / R a car on a banked track or a banking plane can be tricky but just remember that it's different from a normal mass on a slope situation the main force here is the perpendicular reaction force of the track or the up thrust of the plane which is at an angle these forces need to of course supply a vertical component that is equal to the weight mg usually we need to find out what the centropedal force is which is pointing towards the center of the circular track or the plane circular path drawing a triangle shows that this horizontal force is equal to mg tan theta as we have the opposite and adjacent sides this is then equal to mv^ 2 / r and you can see that the m's cancel too simple harmonic motion describes any object that is oscillating around a point like a pendulum or a mass on a spring there are two conditions the acceleration must be proportional to the object's displacement from equilibrium and it must be in the opposite direction to the displacement what's true for acceleration is also true for the force it's trying to return the object to equilibrium hence why we call it the restoring force the equation a= minus omega^ 2 x shows both of these conditions as we know from circular motion omega is equal to 2 pi f well 2 pi /t and x is the displacement from equilibrium in meters if we use calculus in a level physics we derive this from the other equation you'll see in a minute but as we don't we can just use it as our jumping off point for SHM in calculations we're never really concerned with the minus as we know is going to change from negative to positive as it oscillates the maximum acceleration occurs at the maximum displacement which you know is the amplitude capital a so little a max equals omega^ 2 big a we've just swapped out displacement for amplitude if you draw a graph of how displacement changes with time for something undergoing shm like a pendulum we end up with a sign graph to be clear it's sign if it starts at zero cos if it starts at amplitude many people don't realize that these are the same thing just shifted 90° at any time the displacement x is equal to a fraction of the amplitude that fraction is given by s or cos omega t be careful when using s or cos with any periodic motion like circular or shm your calculator must be in radian mode the full equation for velocity at any point is this plus or minus omega * a^ 2 - x^ 2 luckily it's highly unlikely you'll have to use this as more often than not we're just concerned with the maximum velocity which it has at equilibrium where x equals zero so that means vmax equals omega a or 2 pi fa this is an important equation because it allows us to find the maximum kinetic energy if we use this speed in half mv^ 2 and the maximum kinetic energy is equal to the total energy of the system going back to our graph quickly this is essentially distance against time so if you're asked to find the maximum speed from it you could draw a tangent at equilibrium and you'll get the maximum gradient but the far more accurate method is to get the time period and amplitude from the graph then calculate vax= 2 pi fa which is the same as 2 pi a / t as the object moves away from equilibrium kinetic energy is turned into potential energy in the case of a pendulum that's gravitational potential energy of course be careful with displacement for a pendulum though the displacement is roughly a straight line from equilibrium to where it is it's not the vertical height it's moved that means that if you're given the maximum height of a pendulum the only thing you can do with that is find the GPE and equate that to kinetic energy in order to find Vmax height is not amplitude for a pendulum you can then find out more about the oscillations using Vmax= 2 pi FA if there is no damping or resonance and it's a closed system no energy should be lost to the surroundings which means the total energy stays the same as kinetic energy and potential energy do their balancing act in reality there are resistive forces like air resistance that oppose the motion of the object moving it always acts in the opposite direction to the velocity it's a damping force it removes energy from the system if we draw what happens to oscillations over time we can see that the amplitude decreases gradually this is what we call light damping sometimes you'll see a line that join the peaks together which shows us exactly how the amplitude is changing over time you could be asked to state how much energy is left at a certain time in that case just remember that Vmax is proportional to the amplitude and energy is proportional to V^ squ so that means that energy is proportional to amplitude squared so if the amplitude has halved that means the energy will have quarted if the resistive force is a lot larger the object won't even get a chance to oscillate its displacement will just gradually decrease this is like if you grab someone on a swing to let them down slowly we call this heavy damping or over damping critical damping is when the damping force is applied just as the object reaches equilibrium ensuring that it comes to a stop at equilibrium as quickly as possible this would be like you grabbing someone on a swing as they reach the bottom shock absorbers in car suspension are designed to do this as much as possible so the wheels move upwards when going over a bump but go straight back down to where they're supposed to be equilibrium so the car is affected as little as possible resonance is the opposite it's when an external force is driving the oscillations increasing the amplitude adding energy into the system we say this only happens when a system is likely damped maximum resonance happens when the frequency of the driving force matches the natural or resonant frequency of the system that's the frequency it would oscillate at without the driving force the driving force frequency can also be a multiple of the natural frequency and you'll still get substantial resonance when resonance occurs the driving force and the restoring force are 90 degrees out of phase with each other which means the driving force pushes most when the object is at equilibrium much like the wheel under a swinging pirate ship ride pushing it higher and higher every swing if you draw a graph of amplitude against driving frequency f we get the highest amplitude the most resonance when it's equal to the natural or resonant frequency f0 if you add more damping the height of that peak decreases as you reduce the effect of resonance note that the peak doesn't get any narrower though if you have a block of ice and supply heat to it its temperature will increase the particles vibrate faster which means they gaining kinetic energy however once it reaches the melting point of 0° C its temperature will remain constant until it's all melted only then will its temperature start increasing again the same thing happens when it reaches 100° and it turns into a gas but why the constant temperature after all energy is still going into the ice but during a change of state the particles don't gain kinetic energy but rather potential energy an increase in temperature means we must have an increase in kinetic energy of the particles while a change in state when heating anyway must mean an increase in potential energy during a change of state the temperature stays constant so we can't use the specific heat capacity equation instead we use the SLH equation specific latent heat slh of a substance tells you how much energy is needed to change the state of 1 kilogram of it you know all of this from GCSE to be honest but the questions you can get can be fairly complicated at a level for example you might get two substances at different temperatures touching and they reach a common temperature in that case you equate the SHC equations for both but replace delta T with T minus T1 for the colder substance or object and T2 minus T for the hotter one then expand multiply out and rearrange to find the common temperature T if it's an ice cube melting in a drink for example you must add on the SHC equation while it's ice and the SLH equation there are many variations on these questions so just remember you equate the total energy gained by one object to the total energy lost by the other the Celsius scale is not an absolute scale that is it doesn't start at zero we know that the higher the temperature the higher the kinetic energy so we must have a scale that starts at zero when particles have in theory zero kinetic energy that's the Kelvin scale same increments as Celsius just shifted by 273 minus 273° C is 0 Kelvin this is absolute zero to convert degrees C to Kelvin just add 273 particles can be cooled to almost absolute zero but even if you don't have the kit to do this we can still cool a gas and measure the change in pressure at a constant volume or change the volume at a constant pressure plotting pressure or volume against temperature in degrees Celsius we end up with a straight line extrapolating this to where it meets the x-axis a pressure of zero is absolute zero doing this by hand isn't very accurate so it's better to do this algebraically find the gradient then pick a point y1x1 and then we just say y = mx + c so that means that y1 - mx1 = c and so therefore that's also equal to y2 mx2 and you're looking for x2 when y2 equals zero a pressure or volume of zero you must always use Kelvin when dealing with calculations involving temperature the only exception is the SHC equation because that's a change in temperature a change in degrees Celsius is the same as a change in Kelvin this is Boil's law pressure is inversely proportional to volume at a constant temperature so P1 V1 equals P2 V2 we call this an isoothermal change iso means same charles's law is this volume and temperature are proportional at a constant pressure so V_sub_1 over T1 equals V2 over T2 the pressure law also known as the Galac law pressure is proportional to temperature at a constant volume combining these three laws we can say that PV is proportional to T to turn this into an equation we need a constant that's big N * K where big N is the number of gas particles or molecules and K is the Boltzman constant as big N is huge and K is tiny we can swap them out for little N thus moles and R the molar gas constant or we might just say gas constant which is 8.31 much nicer numbers to deal with if only two out of PV and T are being changed we just use one of the above laws if all three are being changed and there's no molecules in or out so N or N stays constant then P1 V1 over T1 equals P2 V2 over T2 we explain gas pressure with the kinetic theory of particles but our model only works perfectly with a theoretical ideal gas some real gases come close like ironically named perfect gases we must make five assumptions for an ideal gas so we can then use our laws it's the acronym raved r is for random particles move randomly we know this from Brownian motion demonstrated by the fact that if you put smoke particles under a microscope you can see them moving randomly due to air particles colliding with them a is for attraction as in there is no attraction between the particles of the gas v for volume the volume of the particles is negligible compared to the volume of the container e for elastic all collisions are elastic and D is for duration the duration of every collision is negligible compared to the time between collisions to derive the kinetic theory equation we imagine a cube box of sides lx l y and lz a molecule inside collides with one of the sides the area ly * lz and it bounces off its change in momentum is -2 mu but we can drop the minus we know that force equals change in momentum over time but it's not exerting this force all the time so we use the time it takes between these collisions it takes two lots of distance divided by speed so 2 lx / u therefore force is equal to 2 mu / 2 lx / u which gives us mu ^2 / lx pressure is force divided by area so dividing by l y lz we end up with p = m u ^2 over all three lengths and so that's the same as mu ^2 over the volume as any molecule can travel in three dimensions we say that u ^2 is equal to a3 cr squ crms is the root mean square speed this is essentially a usable average speed of the particles in a gas we know that in reality the average velocity is zero as the particles travel in all directions equally when averaged to calculate U RMS that's just the RMS value in one direction we add up their squares divide by the number of molecules then square root and so then C is a combination of U V and W the velocities in all three directions multiplying this equation by the number of molecules big n gives us the pressure of the whole gas rearranging we can see that pv is equal to a3 nm mc rms squ remember m is the mass of one molecule not the mass of the gas however big n * m does give us the total mass of the gas so that's why we can replace n / v with the density of the gas so p is equal to a3 row c^ 2 when doing calculations don't bother writing the RMS keep things simple as PV is also equal to NKT we find that MC ^2= 3 KT half this and we have kinetic energy half MC^ squ C basically being the same as V so the kinetic energy of one molecule in a gas is equal to 3 KT that means if you know the temperature of a gas you can know the energy of one molecule multiply by big N of course and you have the total kinetic energy of the gas notice there is no mass in the equation that means that no matter what particles you have in a gas say it's air it has nitrogen oxygen carbon dioxide molecules they all have the same kinetic energy because they're the same temperature however as they do have different masses they will have different speeds of course you saw what a PV graph looks like for Boil's law earlier we can draw an arrow to show whether the gas is being compressed or expanding at a constant temperature we can draw what happens if a gas is compressed at a constant pressure only volume decreases the area under any line on this graph equals work done newtons per meter squar times by meters cubed gives nm newtons* meter that's jewels work done if a gas is being compressed we say that work is being done on the gas if it's expanding we say the gas is doing work if the pressure is constant we can say that the work done by or on the gas is equal to P delta V if you have a straight line going down that's a decrease in pressure at a constant volume so no work is being done whenever you supply heat to a gas it can cause its temperature and therefore its internal energy to rise or it can cause the gas to expand and do work in most cases it does a combination of both the first law of thermodynamics is therefore simple q equals deltaU or sometimes just U plus W heat into the gas equals change in internal energy plus work done in expanding seems simple but using the equation can be tricky however firstly you need to deduce which of these are positive negative or zero q is positive if heat goes in negative if heat leaves if it's a closed system and no heat enters or leaves then this is an a diabetic change so we can say Q is zero so that means U is equal to minus W we can also say that P1 V1 gamma equals P2 V2 gamma where gamma is the a diabetic constant u is positive if the temperature of the gas increases if it gets hotter it's negative if the temperature decreases u is zero if the temperature doesn't change an isoothermal change if that's the case we can say Q equals W finally W is positive if the gas expands the gas is doing work w is negative if the gas is being compressed work is done on the gas instead if pressure is constant we can replace W with P delta V w is zero if the gas stays at a constant volume so that means Q equals U then all you have to do is pop in your numbers to find one of these that will be an unknown it's believed that stars are the result of dust and gas particles in clouds we call such a cloud a nebula being attracted to each other due to gravity the cloud becomes hotter and more dense until fusion starts to occur a star will remain stable so long as the outward pressure from fusion and the force of gravity pulling inward remain balanced we say it's in the main sequence stage of its life when a star dies the outward pressure increases which causes it to expand turning it into a red giant if it's a star a similar size to our sun or a super red giant for stars much bigger than our sun a red giant will then collapse once all the fuel for fusion has run out leaving a white dwarf and then a black dwarf once it's cooled a super red giant explodes we say has gone supernova leaving a very dense neutron star at the center or an even more dense body that causes a black hole there the outer layers of the supernova move away forming new nebula from which new stars could be made nuclei fuse together to make heavier elements some of these could only be made as a result of the huge amount of energy released from a supernova luminosity of a star is just the term we give to its total power output in watts it's given by Stefan's law l in watt equals the Stefan constant time the area of the star so 4 r² time it surface temperature in Kelvin to the power of 4 in reality the specific wavelengths emitted by a star is dependent on a lot of things but for the sake of physics we can treat stars as black bodies we assume that they emit and absorb all wavelengths of light that allows us to draw this graph of energy emitted against wavelength we can see that it increases with a shorter wavelength because well shorter wavelength means higher frequency which means that the waves have more energy but it does reach a peak and then comes back down according to classical physics this energy should carry on increasing with shorter wavelength but this implies that a star has infinite energy to give out which is impossible and that's because of some quantum mechanics malarkey as it happens at around ultraviolet wavelengths this is called the UV catastrophe the wavelength it emits with the highest energy we call lambda peak the hotter the temperature the shorter this wavelength is and the higher the peak this peak wavelength is given by V's law wavelength time temperature in Kelvin equals V's constant which is 0.0 0029 we can draw a graph of luminosity against temperature to show what stars exist out there just remember that temperature decreases as you go along the x-axis main sequence styles follow this line here showing that for stars in the middle of their life cycle luminosity follows temperature which is what we would expect white dwarves are small but hot while red giants are huge and relatively cool so they occupy the spaces above and below the line astronomical distances are so large that we usually use the parseek as our unit instead of meters it's short for parallax second this second isn't a time though but it's a 60th of a 60th of a degree which is what we call an arcsec one parseek is the distance at which one au subtens 1 arcsec 1 au being astronomical unit which is the mean radius of the earth's orbit around the sun we can also use the lightyear instead that's the distance light travels in one year we have a good idea of what wavelengths of light are emitted from stars however when we look at distant stars and galaxies these wavelengths appear longer they're shifted towards the red end of the spectrum the light has been redshifted much like when the pitch of an ambulance siren drops when it's moving away from you like this shows the galaxies must be moving away from us and this is the case in every direction we look not only that the light from more distant galaxies is even more redshifted suggesting they're moving faster away from us receding faster we say at a faster rate this implies that if we go back in time all of these galaxies appear to have originated from the same point in space this is therefore used as evidence for the Big Bang theory and that the observable universe is still expanding the other piece of evidence for it is CNBR cosmic microwave background radiation outer space might look dark but we can detect microwave radiation being emitted from very far away from all directions this could be emitted as a result of matter still cooling down so it seems we're looking at the edge of the big bang which is essentially still going we can give the amount of red shift or Doppler shift a number this is equal to the recessional speed divided by the speed of light and it so happens it's pretty much equal to the change in frequency or wavelength divided by the original values this approximation only works if the speed is much less than the speed of light which of course is true for galaxies the shift in wavelength can be seen on the emission spectrum from a star characteristic wavelengths we observe emitted from gases here on Earth are generally shifted to the longer end sometimes a line will appear to be split in two this can be due to a binary star system two stars that are trying to orbit around each other so end up just orbiting around a common point between them one could be moving away from us while the other is moving towards us so we have a different red shift for both producing two separate lines twice every orbit the line recombines into one of course as their speeds relative to us will be the same at those points the Hubble graph is just a graph of the recessional velocities of all the observable galaxies against their distance they're proportional which is why the Big Bang Theory is widely accepted making an equation we say that v equ= h0d where h0 is the hubble constant the gradient of the graph this has the unit seconds to the minus1 therefore the reciprocal of this gives us seconds the value of which in theory is the age of the universe however often the graph won't be in me/s and meters it'll be in kilm/s and mega parex if you're asked to calculate h0 in seconds to the minus1 then you must convert these first we say that objects can experience a non-cont force if they're in a field gravitational fields and electric fields work almost identically it's just that gravitational fields affect all objects while electric fields only affect objects that have a charge you already know that gravitational field strength G has the unit newtons per kilogram it tells us how much the force of gravity is pulling on each kilogram of mass an object has on Earth this is 9.81 but the higher you go the weaker the field gets this is shown if we draw the field lines around the earth these show the direction of force and you can see they diverge the closer the field lines the stronger the field this is a radial field if field lines are parallel however the field strength is the same everywhere we call this a uniform field if you drew the gravitational field lines in the room you're in now they'd appear pretty much parallel so we can say that on a small scale gravitational fields are uniform electric fields are slightly different too because they can also repel objects depending on their charge for consistency electric field lines always show the direction of force on a positive charge placed in the field that means around a positive sphere say a vandagramraph generator for example the field lines point away from it as a positive charge would be repelled if you put it in the field if you put a negative charge in there it would be attracted to the sphere for a radial field gravitational field strength is proportional to the mass creating the field and inversely proportional to the distance from its center squared the constant we need to turn it into an equation we call the universal gravitational constant big G this is 6.67 * 10us 11 we give electric field strength the symbol E don't get it confused with energy the constant here is 1 / 4 piepsilon0 epsilon0 being the permitivity of free space which you'll find on your formula sheet however calculating 1 / 4 pi epsilon0 gives us 8.99 * 10 9 so we can just say that this constant altogether is 9 * 10 9 as a shortcut and we can just call it k you'll get the correct answer if you use this and it saves a lot of time and hassle so I'll be using it from here on out we can draw the inverse square relationship for G and R on a graph but for a planet G field strength is at a maximum at the surface if you dig downwards towards the center of the earth g decreases linearly to zero at the center we can prove this actually we know g is proportional to m / r squared density however is proportional to m / r cubed as it's equal to mass over volume that means g is proportional to row r if row is constant which we assume it is inside of a planet g is just proportional to r hence our linear relationship inside the planet this proportional relationship is also useful when comparing field strengths on the surface of planets with different sizes and densities if you think about it it makes sense because if you're at the center of the Earth you'll be pulled in all directions equally so there would be no overall field or force that's because field strength much like force is a vector quite often you'll be presented with two planets and you'll be asked to find out where there would be no overall force or field in between in that case we say G1= G2 for the two planets that means big GM1 / R1^ 2 equals GM2 / R2 where R1 and R2 are the distances from the center of the planets to the point you could also be given the overall distance D instead in this case we replace R2 say with D minus R1 you'll then rearrange for R1 by swapping M1 and D minus R1 so that ends up on top and then we just expand so we end up with a minus1 and then rearrange to finish it off as G has the units Newtons per kilogram and electric field strength Newtons per kum to find the actual force a field exerts on an object we only need to multiply by the mass or charge that we're actually putting in the field so F= big G mm / R 2 you might say M1 M2 instead you might use D as well this is Newton's law of gravitation for electric fields f= KQ / R 2 this is Kulum's law remember r must be the distance between the center of the two objects you could be asked to define these two laws using your words newton's law of gravitation is defined like this and we can just use the equation just take out actual SI units the gravitational force between two objects is proportional to the product of their masses that just means times together and inversely proportional to the square of their separation for Kulum's law we just swap out masses for charges instead you know that F= mg for gravity as that's the same thing as weight well it's also F= ma as well isn't it because we know that gravitational field strength is also the same as acceleration due to gravity for electric fields F= EQ this is a very simple equation that's always used but it's one people often forget you know the work done is equal to force times distance so that's F * R in this case therefore if an object is pulled away from an object that's attracting it like a rocket putting a satellite in orbit then it must have gained potential energy that means GPE is equal to GMM / R an electric potential energy KQ over R you might say "But I thought it was mgh." That's only if the field strength stays the same you're in a uniform gravitational field like we said on a small scale we can say that's the case but not when we're looking at the big picture much like we've seen newtons and newtons per kilogram we have jewels so now we need jewels per kilogram this is called potential symbol V the definition is this the work required to move unit mass from infinity to that point the reason we use infinity as our baseline is because that's the only thing that's common to every object in the universe so we say that's where you have a potential of zero as an object gets closer to a planet the potential gets more negative potential is a scalar so it's the same in all directions that means we can draw equipotentials around a planet dotted lines that show certain potentials much like contours on a map the closer together these are the stronger the field no work is required to move an object along an echo potential that's why satellites will orbit a planet without any help electric potential is measured in jewels per kum instead of course and we just replace unit mass with unit positive charge instead for gravitational potential V equals minus GM / R electric potential V equals just KQ / R no minus is needed because if we have an attractive field like a planet well we know that it must be a negative charge creating it because it has to be attractive to a positive charge everything's done on if we put a positive charge in an electric field of course if it's a positive charge creating the field instead you'll end up with positive potentials gravitational potential can only ever be zero at infinity whereas electric potential can be zero between oppositely charged objects as it's a scalar you just sum the individual potentials at a point to find the overall potential there just remember that between two like charges field strength can be zero but potential will not as V is proportional to M / R and density is proportional to M / R cubed V is proportional to row R 2 if you have a graph of V against R there are two things we can do with it we can read off two points calculate the change in potential in jew per kg or jew per kum then multiply by the mass or charge of the object we're actually moving to find the energy or work done so that means change in potential energy is equal to m delta v or q delta v it might be tempting to put the distance moved as r in our equation but remember r is always the distance from the center of the mass or charge that's producing the field that means that if we want to calculate change in potential instead of using a graph we must do KQ / R1 minus KQ / R2 of course you can then factoriize this however if you find the gradient of a potential distance graph with a tangent this is equal to the field strength at that point that's because going full circle with our equations you can see that V / R gives us the equations for field strength that's why the alternative name for field strength is potential gradient we can also just do delta V over delta R to find it if we've been given two potentials and the distance between them we know from circular motion that centripal force is equal to mv^2 / r for a satellite this is due to gravity and so that's gm m / r 2 we can equate these little m's and an r cancel and we find the speed of a satellite is equal to roo gm / r if you're dealing with a time period instead of speed though gm / r 2 is equal to m omega^ 2 r instead and we can replace omega with 2 p / t where t is the period of orbit we can flip this and see that t ^2 is proportional to r cubed this is Kepler's law well his third law to be precise it means if the satellite gets closer to the planet say its time period decreases if you're asked to find out how t or r specifically change you merely divide one power by the other also you should know that the speed increases as v equals root gm / r and therefore its kinetic energy increases too a satellite in a geocynchronous or geostationary orbit has a time period the same as that of the Earth's rotation 24 hours quite often you'll have to put this into an equation but it must be in seconds so multiply by 3,600 the only place it can stay above the same point on the Earth is at the equator that means there's a ring of satellites at the same height above the equator these are used for communication GPS television and internet a low polar orbit is one which is lower and according to Kepler's law they have a shorter time period they don't stay above the same point as they move from pole to pole sweeping out different parts of the Earth as it rotates underneath these are used for surveillance and weather just be careful if you're given the height of a satellite's orbit above the surface you need to add on the radius of the planet to get that proper R you can sometimes be asked the question how fast would the Earth need to spin for you to appear weightless in that case you've just turned into a satellite and your R is the same as the Earth's radius all we do is use m omega^ 2 r again from there we just do exactly the same thing as we've done above if you want to escape a gravitational field you must be given enough kinetic energy at the surface to be equal to the potential energy needed to get to infinity to a potential of zero that means half mv ^2= gmm / r where v is the velocity you need to fire something at from the surface and r is the planet's radius the little m's cancel and we get v= 2gm / r so that goes to show that every object regardless of mass has the same escape velocity at the surface of a planet you might notice that we have GM over R which is the same thing as potential so actually yes we can say that V escape velocity equals roo 2V where big V is the potential at the surface just don't get confused between your V's distance of closest approach is the same idea but in reverse if you have a positively charged particle moving towards another it will be repelled if moving directly towards it the minimum distance it gets to before it turns around and comes back the way it came is called the distance of closest approach at this point it has a speed of zero much like when you throw a ball up in the air and it reaches the apex so again K equals P so half MV^ 2 equals in this case KQ / R where R is our distance of closest approach you make a uniform electric field if you have two charged parallel plates the field lines are parallel and go from the positive plate to the negative plate again the field strength is the same everywhere if you put an electron in here for example it will accelerate towards the positive plate with a constant acceleration because the force is constant because the field strength is constant for a uniform field like our parallel plates the equipotentials run perpendicular to the field lines in fact field lines and equipotentials always cross at 90° to each other no matter what shape the field as these equipotentials are equally spaced that again shows that the field strength is the same everywhere we know that electric potential is measured in jewels per kum but we know that that's the same as volts that means if we take delta V over delta R and just replace them with the voltage PD between the two plates and the separation of the two plates for R we find that electric field strength is equal to just V over D if a charged particle enters the field at a right angle its trajectory is parabolic and it's then identical to a normal projectile motion question you use suvat for one component of its velocity and your speed equals distance over time for the other component as that doesn't change so a= eq over m or vq over md replacing electric field strength if our parallel plates are making an object levitate for example like in Milikin's oil drop experiment we can say that the force due to gravity and the force due to the electric field are equal and opposite so mg equals eq replacing e with v / d we find the charge of the object is equal to mgd over v electricity is the flow of charge or charges like electrons they carry energy from a source of energy to a component by the way you're going to see me mix up cells and batteries in this video because they're just the same thing really and they do the same job the battery has a store of chemical potential energy when connected in a complete circuit this energy is transferred to the electrons which moves through the wires this movement of charge is called a current and we say it always goes from the positive terminal of the battery to the negative don't think about it too much as the electrons pass through the bulb their energy is converted into light but the electrons don't just disappear once they transfer all the energy to the bulb as this is one big loop these electrons are pushed back round to the battery by the ones behind them where they're refilled with energy ready for another trip around the circuit this constant flow of electrons transferring energy is what keeps the light bulb on because electrons are so small and there are so darn many of them we don't deal with individual electrons but instead deal in kulum of electrons or kulum of charge potential difference PD for short also known as voltage tells us how much energy is transferred per kulum of electrons so if a cell or battery says it's 1 volt that means that one jewel of energy is given to every kulum of electrons that pass through it if a battery is 6 V that means 6 Jew is supplied per kum instead we measure PD with a voltmeter they're always connected in parallel to the components you want to measure the voltage of in the real world that means the leads or cables from the voltmeter always peg back into other leads if we put the voltmeter across the battery it should measure 6 volts right because 6 V is supplied to the electrons in the circuit that's just 6 jewels per kum but put it across the bulb and it should still say 6 V why because the electrons have to lose all of that 6 V worth of energy as they pass through here's the equation for PD pd in volts is equal to energy in jewels divided by charge in kulum in simple form V is equal to E over or divided by Q q is the symbol for charge but it's measured in C in kulums you'll see the rearranged version E= QV on your formula sheet current on the other hand tells us what the rate of flow of charge is like any equation for a rate as per usual it's something divided by time so here it's current in amps equals charge in kulum divided by time in seconds or i= q / t yes we use capital i as the symbol for current not see blame the French for that as they call current intensitated kant it does mean though that we don't get confused between current and kulums though so we stick with it you're going to see the rearranged version of this equation on your formula sheet q= I T that's I * T we measure current with an amter note that it's not ampmeter unlike a voltmeter it must go in series that means in line with the component we want to measure the current for components in a circuit have resistance that is they resist the flow of charge or current through them but that's not a bad thing this has to happen in order for them to work a bulb has resistance which causes energy to be transferred and light to be emitted a resistor of course has resistance too but it just produces heat when current flows through it if we make a circuit with a resistor and change the PD available to it what we find is that an increasing PD results in a greater current flowing in fact doubling one doubles the other so we can say that PD and current or V and I are directly proportional drawing a graph of these two makes a straight line and if we turn the battery round we can get negative values for both two but still a straight line through the origin this straight line a constant gradient shows that a resistor has constant resistance we say it's omic the steeper the gradient of this line the lower the resistance of the resistor as more current is flowing per volt the equation for resistance is Ohm's law v= IIR we can get the resistance of a component from an IV graph like this by just picking a point on the line and rearranging Ohm's law so R is equal to V / I for a resistor you'll end up with the same answer no matter what point you pick if you repeat the same experiment for a bulb in place of the resistor however you'll end up with a curved graph like this this shows that the resistance is changing the resistance of the metal filament in the bulb in fact you'll find that any metal has a changing resistance if you increase the PD and current they're nonomic at higher PDS the current increases less and less so that means they can't be proportional this shows that the resistance of the metal is increasing with a higher PD and higher current the change in gradient shows us that this is true but we still just take a point on the line and use Ohm's law if we want to find the resistance it's just that it does matter where you pick that point in this case so why does resistance change for a metal well it's because metals consist of a latis or grid of ions surrounded by a sea of deoized electrons that just means they're free and free to move or rather they're fairly free to move because they do collide with the ions as they flow that's why the metal heats up when you pass a current through it the higher the current the more frequent these collisions are and this makes the ions vibrate more and more which in turn makes it harder for the electrons to flow the resistance has increased now there is another component called a diode it will give you this graph the circus symbol might give you a clue as to why this is a diode only lets current flow through in one direction we say that in one direction the resistance is very high and it's very low in the other which is why the current increases suddenly at around 1 volt an LED is a light emmitting diode similar symbol just with a couple of extra bits these are what most lights in electronics are these days rather than filament lamps they act in the same way as a diode so they give you the same graph but they just happen to emit light as well a superconductor is a material that has a resistance of zero not nearly zero zero generally these materials need to be cooled before they reach that point so you might see a graph like this the temperature is going down and so the resistance goes down until boom we hit what we call the critical temperature at the critical temperature and below resistance falls to zero every material has a different resistivity or resistivity some people say the definition is this it's the resistance of a cube of unit length sides of that material so in SI units that's the resistance in ohms of a 1 m cube note that that is not resistance per meter cubed the definition must involve an actual cube the unit we use is ohm meters again be careful that's not ohms per meter to find resistivity we measure the diameter of a wire with a micrometer and calculate area from that using pi d^2 over 4 we then measure V and I to calculate R for varying lengths L of the wire measured with a meter rule plotting R against L gives a proportional relationship the equation for resistivity therefore is this r= row L / A where row is resistivity in ohm meters rearranging this we have row equals RA A / L the gradient of the graph is R / L so multiplying by the area gives us the resistivity series and parallel circuits this is where things get a bit tricky remembering what happens to current PD and resistance when we have components in series or in parallel here's the simplest series circuit we can make really just two resistors in line with the battery what you need to remember is that for components in series total PD is shared between them current is the same for all of them and the total resistance is just the sum of all resistances that just means added up let's deal with that first point if these resistors are the same let's say 10 ohms each then that 6V total PD from the battery must be shared between them so if we put a voltmeter across each of these they'd both read 3 volts it wouldn't matter what resistance these resistors are they could be a million ohms each if they're the same then that total PD is shared equally by the time the electrons leave the second resistor they have to have lost all 6 volts worth of energy ready to go back to the battery to be refilled this idea is actually just Kirchoff or Kiaov's second law the sum of emfs must equal the sum of PD drops in a closed loop just remember that if batteries are pointing in opposite directions one of them must be a minus by the way we can also call this setup a potential divider circuit as the total potential total PD is being shared if the resistors don't have the same resistance then we can use the second point to help us that is the current is the same for both let's say the first resistor is 20 ohms using 4 volts of the total 6 volts available we know two things out of VI and R so let's use Ohm's law to find out the third for it current in this case I rearranging Ohms law we get I is equal to V / R so that's 4 divided by 20 0.2 amps same for the second resistor too is there also a second thing we know about the other resistor why yes there is remembering the first rule up here we know that if the first resistor is using 4 volts of the total 6 Vts available well then the other resistor must be using up 2 V we could then use Ohm's law again to find its resistance 10 ohms the rule of thumb is this the greater the resistance the greater the share of the total PD it gets we can also use Ohm's law for a whole circuit we just need to make sure that we're dealing with the total PD total current and total resistance the rules for parallel circuits are the opposite the PD is the same for every branch again this is true because of Kikov's second law the battery is actually involved in two loops so therefore the PD drops in both loops must be the same current is shared between each branch and the more resistors you add in parallel the lower the total resistance this by the way is because you're giving the current more roots to move through the circuit which means more current can flow so if these two resistors are connected to the 6volt battery in parallel you know straight away that the PD for both has to be 6 V voltage isn't shared in parallel circuits if however we say 0.5 amps total current is flowing through the battery and 0.2 amps of that is flowing through the top resistor that must mean that there's 0.3 amps flowing through the bottom resistor this is actually Kikov's first law current and therefore charge must be conserved at any junction if you're not in a rush why not pause the video and see if you can calculate these two resistances by the way if you want a little bit more help on this then have a look at my video how to answer any electricity question it's not only metals that can change resistance we can have a thermister and you have a circuit that responds to changes in temperature a thermister's resistance decreases if the temperature increases so in essence it does the opposite to a metal by the way you might see a thermister called an NTC negative temperature coefficient that just means that higher temperature lower resistance in this case if the temperature increased the resistance of the thermister would go down as does its share of the total PD that means the PD measured by this voltmeter will increase this could be the basis of a temperature sensor for your central heating for example an LDR is a light dependent resistor very similar to a thermister but resistance goes down with increased light intensity not temperature so this circuit could be on the top of a street lamp light intensity goes down resistance of the LDR goes up as does its share of the voltage this could then be connected in some way to the light bulb so it turns on as it gets dark power is the rate of energy transferred so energy divided by time however when it comes to electricity we can also calculate it with this equation too p equals VI power equals voltage PD times current a battery or cell produces DC direct current the PD and therefore current also only point in one direction electronics need DC to work however AC alternating current is needed to transmit electricity over long distances say through the national grid this is because transformers are needed to step up the voltage reducing the current before it enters the grid more on transformers in magnetic fields the neutral wire in main stays at a potential of 0 volts while the live wire varies between plus 325 volts and minus 325 volts so we say the peak voltage is 325 volts and we might say that the peakto peak voltage is double that 650 however to do any measurements with AC we must convert it to a DC equivalent by using root mean square values to convert from peak voltage to RMS voltage or VRMS the conversion factor is 2 325 divided by roo2 gives us 230 volts which you know is what UK mains voltage is known to be it's the RMS value that we're given and it also works going from peak current to RMS current to go from peak power to mean or average power it's a little bit different we don't call it RMS power by the way it's not quite the same as peak power equals peak voltage time peak current we divide by V and I by 2 to get both RMS values doing that twice is the same as dividing by two so that means that the average power is half the peak power delivered it turns out batteries or cells have a resistance of their own so if you attach a bulb to a 6V battery the bulb will get less than that a voltmeter across it might measure 5.5 volt say this would also be the same if we attached the voltmeter across the battery terminals instead so we can call this the terminal PD be careful that means the voltage available to the rest of the circuit that means 0.5 volt is being lost inside the battery due to its internal resistance little R the emf electromotive force epsilon is the symbol we use is the total PD provided that's the 6 volts here so the equation is this emf equals terminal PD V plus I little R where little R is internal resistance so I * little R is the voltage loss due to internal resistance if we increase the load resistance that's the resistance of the circuit the current flowing through the battery decreases of course but this results in less PD being lost in the battery so the terminal PD increases drawing a graph of terminal PD against current gives us a straight line the magnitude of this gradient is equal to the internal resistance if we extrapolate the line back to the y-axis the y intercept is equal to the emf which makes sense as if there's no current flowing then in theory the circuit should get the whole of the emf there's no volts lost in the battery you can of course also get the emf by just attaching a voltmeter across the battery by itself with nothing else connected this works as voltmeters generally have very high resistance the reason that thermisters and LDRs work is because they're made out of semiconducting material such materials have electrons that are only free to move when they are provided with enough energy to first move from what we call the veilance band to the conduction band passing straight through the forbidden band semiconductors sit between insulators and conductors you might have guessed but it's a spectrum naturally we can tell how well a material conducts electricity by comparing their number densities that is how many charge carriers not necessarily electrons there are per meter cubed for conductors like metals the number density is around 10 to the 28 semiconductors around 10 17 give or take moving more electrons into the conduction band changes this number of course insulators have very low number densities drift velocity is what we call the literal speed of an electron as it flows through the wire in meters/s i don't see the point in knowing this but there we go the equation is this current is equal to cross-sectional area time the number density time charge of an electron time drift velocity we can then rearrange this for V if needed a capacitor is a device that stores charge and therefore energy if a PD is applied across it double V and Q doubles they are proportional the gradient of this graph is the capacitance of the capacitor which tells you how many kums of charge are stored per volt this is the unit farad the equation is therefore Q= VC as energy is charge times voltage that means the area under the graph gives us the energy stored in the capacitor as it's a triangle the equation is E=/ QV substituting out Q or V with our first equation we get the alternatives/ CV ^ 2 and half Q ^2 / C you can't really measure charge directly which is why we have to calculate it using charge equals current time but this only works however if the charging current is constant as the current will start high and fall to zero if we just hook up the capacitor up to a battery we need a variable resistor that we're constantly changing in order to keep the current constant once fully charged the PD across the capacitor will be the same as the batteries it's worth remembering that before this point the rest of the PD from the battery will be taken up by the resistor in the circuit when discharging a capacitor usually through a resistor the PD decreases or decays exponentially similar to nuclear decay the equation is this v over V 0 equals E the minus T / RC where V is the PD at time T now and V 0 is the initial PD t is the time R is the resistance we're discharging through and C is the capacitance you might see it as V equals V 0 etc but this way you can see that it's just a ratio which will be between 0 and 1 how much of the initial voltage do we have left it also works for current I and charge Q as well thinking about it being a ratio that makes it easier to rejig this for charging instead we just want the other proportion so it's equal to 1 minus e to the minus t / rc where v 0 is the maximum final pd when fully charged in this case again this works for charge but current still decays when charging it doesn't increase it decreases to rearrange for tr or c we natural log both sides leaving us with a log yes I can say that even though it's a natural log log v over v 0= minus t / rc this is needed when doing an experiment to find an unknown capacitance we measure v as it changes with time as it discharges then We can either do a graph of minus log v over v 0 against t and the gradient will be 1 / rc or we can use our log identity to say that log v over v 0 is equal to log v minus log v 0 we plot log v against time and again the magnitude of the gradient is equal to 1 / rc we use halflife to compare radioactive isotopes but for capacitors we use time constant this is the time it takes such that the fraction of PD left is e to the minus1 which is equal to 0.37 37% of course that can only be true when the time t is equal to RC so t over rc is equal to 1 so the resistance time capacitance is equal to the time constant if you're not in a rush try and prove that the units check out if you're given a decay graph you find 37% of an initial value it doesn't matter when you start actually but the time it takes to do that is the time constant and that's equal to RC the basic structure of a capacitor is two parallel plates which become oppositeely charged when a PD is applied remember no charge should flow directly between the plates in reality these plates or foils are wound into cylinders as you probably see from the designs of the ones you use in the lab capacitance is proportional to the area of these plates and inversely proportional to their separation the constant that makes this an equation is permitivity epsilon this in essence is a measure of how easy it is for an electric field to be produced in the space between if there's air or a vacuum between this is the permitivity of free space epsilon0 you're given the value in your formula sheet if there's anything else in between the permitivity will be greater than that of free space so we say that the permitivity is equal to that of free space times the relative permitivity of the material which is just a number that's greater than one if it's two then that means it's twice as good as free space we can increase the capacitance by inserting a dialectric between the plates an insulating material that contains polar molecules that is they're affected by electric fields we can represent them with just a little peanut shape with a delta plus and delta minus to show that they line up along these field lines as a result the edges of the dialectrical will become charged oppositeely to the plate it's near this increases the capacitance as you can see from the equation another way of thinking about it is that the polar molecules produce their own counter electric field reducing the net field between the plates and therefore that reduces the PD needed to store the same amount of charge if you change any aspect of a capacitor A D or epsilon R you can be asked what the knock-on effects are for V C Q or energy first you need to know if the capacitor is still connected to the battery or not if it is V will stay constant and Q will change if it's disconnected Q is constant and V changes instead it's a bit more complicated with energy as we have three equations to choose from you choose one that has a constant in it so if disconnected we choose E=/ Q ^2 / C this example here can also be explained in terms of electrostatic potential energy like we said the edges of the dialectric are charged oppositeely to the plates which means they're attracted to them to remove the dialectric we must do work to separate them you have to put in energy and where does that energy go it goes into the capacitor all waves transfer energy without transferring matter longitudinal waves are those in which the direction of the oscillations is parallel to the direction of energy transfer that is the direction the wave is going in these waves particles bunch up we call those compressions and when they're spread out we call those rare refractions transverse waves are those in which the direction of oscillations is perpendicular to the direction of energy transfer we can represent any wave including longitudinal waves like this we call this a waveform displacement is up the y- axis basically just how far the particles have oscillated from their original position and it can be either distance or time on the x-axis the peak of a wave is called the amplitude it's the maximum displacement from equilibrium if it's distance on the x-axis one complete wave here gives you the wavelength we give this the symbol lambda for short but it's measured in meters if it's time on the x-axis instead one complete wave gives you the time period this is the time it takes for one complete wave to pass frequency on the other hand is how many waves pass a point every second and the unit is hertz frequency is equal to 1 over time period f= 1 /t you can often be asked to find frequency from a waveform like this measure the time period then do one divided by that easy the wave equation is this v= F lambda that's wave speed equals frequency time wavelength it's worth remembering that visible light wavelengths vary from around 400 to 750 nanometers or 4 to 7.5 * 10us 7 m with red light having the longest wavelength and blue the shortest the intensity of a wave is proportional to the amplitude squared that means that if the amplitude doubles the intensity quadruples when light waves move from one medium to another say from air to glass they change speed in this case the wave slows down and the wavelength also decreases a change in medium also results in a change in direction this is called refraction that is if it's at an angle to the normal the line we draw perpendicular to the surface if light slows down it moves closer to the normal so that means that the angle of refraction is smaller than the angle of incidence now all of these angles are measured from the normal how much a medium refracts light is determined by its refractive index symbol n it's just a ratio that's equal to the speed of light in a vacuum basically the same for air divided by the speed of light in the medium so it will be a number that's always greater than one if you calculate sin I over sin R for a light ray entering a medium from air this happens to also give us the refractive index the full equation for refraction is Snell's law n1 sin theta1= N2 sin theta 2 you can see that we've just replaced I and R with thetas bear in mind that in reality the refractive index of a material varies with wavelength blue light's refractive index will always be higher than red lights for the same medium that means that it refracts more than red that's why white light under goes dispersion when it passes through a prism let's say the light ray is coming out of the glass block and into air now if we keep increasing this angle of incidence eventually we'll end up with an angle of refraction of 90° that light ray will be going along that boundary along the surface the angle of incidence is now equal to what we call the critical angle and that's going to be different for every medium if we make the angle of incidence even bigger than the critical angle that means that no light is going to be refracted out of the block instead all light is reflected back inside by the way we always get some reflection but now we have total internal reflection want to calculate the critical angle well you can start with Snell's law you call theta1 theta c instead make n2° and we end up with sin theta c= n2 n1 this incidentally is how optic fibers or fiber optics work the basic optive fiber is a glass core surrounded by a protective sheath the issue is that this will have a higher refractive index than the core so trir would not happen if it were not for the extra layer called the cladding that is in between them it has a lower refractive index than the core so trir can occur one of the issues with optic fibers is the fact that light will diverge spread out as it travels down a fiber this means rays can take different paths and therefore different times to reach the other end this is called modal or multipath dispersion and it results in pulse broadening if pulses are stretched too much they start to overlap as the next one has started to arrive before the first one has completed its journey your data is a goner to mitigate this we can do a few things number one make the fiber really thin the thinner it is the less light will diverge number two put repeaters in every so often to retransmit the slightly broadened pulses as brand new ones the final clever idea is to use graded index fibers light emitted from the sun oscillates in all orientations however if you have a polarizing filter made of very small lines it only lets half of the light through only half is transmitted the other half is absorbed this is because it only lets waves of certain orientations through we say that if the lines are vertical waves oscillating vertically are absorbed while horizontally orientated waves are transmitted pass through and vice versa waves between vertical and horizontal are transmitted or absorbed according to their angle it's very complicated in reality but put simply the light is now polarized and the intensity of the light is halfed put another polarizing filter at 90° behind the first and all of the light is absorbed a normal wave is what we call a progressive wave the wave moves while the medium or particles just oscillate around a point say up and down two points one wave apart are said to be in phase they're doing the same thing at any time we say their phase difference is zero or 360° as we can represent wave cycles as circles we can also split circles and therefore cycles into radians instead of degrees there are two pi radians in a circle so again the phase difference of these two points is 2 pi radians like we said that's the same thing as zero points on opposite sides of a wave are completely out of phase or we might say in anti-phase they're half a wave out of phase so their phase difference is 180° or pi radians these two points are a quarter of a wave out of phase so that's a phase difference of 90° or p /2 radians we generally don't give phase differences above 180 degrees pi radians as we can just give the opposite phase difference so 270 degrees is just given as 90° etc any phase difference can be calculated as a fraction times the total in a cycle so we can do the distance between two points divided by the wavelength or time difference between two points divided by the time period depending on what graph we're given then we just multiply by 360 or 2 pi depending on whether we want the phase difference in degrees or radians when two progressive waves traveling in opposite directions meet they undergo superpositioning they are superposed when these waves have identical frequency and wavelength a stationary wave is formed the points at which the resulting wave doesn't move is called a node that's because the displacements are always summing to zero at these points we call this destructive interference so the amplitude is zero at a node no energy is transferred at all other points the resulting wave will oscillate due to constructive interference where the amplitude is a maximum these are called anti-nodes we can see this if we attach a string under tension to a vibration generator the wave is sent down the string and it reflects at the pulley and these two waves interfere the simplest stationary wave looks like this we call this the first harmonic or the fundamental one loop one anti-node and a node at both ends we have half away from the string so that means L equ= lambda / 2 the frequency of this is given by the equation 1 / 2 L of the string time T over mu t being the tension in Newtons which is usually the weight of the masses hanging on the end and mu being the mass per unit length of the string so that's in kilog per meter incidentally root t over mu actually gives you the wave speed double this fundamental frequency and we get the second harmonic l equals lambda in this case third harmonic L equals 1 and a half lambda and so on stationary waves can also be formed in tubes of air at a closed end a node must be formed at an open end an antiode we can represent the stationary wave much like we do for a string just bear in mind that the first harmonic is likely to be different than that of the string because one or two of the ends are probably going to be open unlike progressive waves points on stationary waves can only ever be completely inphase or completely out of phase it just depends on if they're the same side of equilibrium or not that's because two points above equilibrium will reach their own different amplitude at the same time and they'll both reach equilibrium at the same time too even if waves aren't traveling in opposite directions they still interfere when they meet thomas Young put a thin slit next to a candle to first ensure that the light used for the experiment was coherent all of the light was in phase but the definition of coherent is this actually always have a constant phase difference this light then passed through a double slit the result was bright and dark fringes appearing on the screen call these maxima and minima we have a central maximum because those two rays meeting have traveled the same distance so therefore they interfere constructively the dark fringes or minima are due to the light from the two slits traveling slightly different distances to each other we say their path difference is half lambda that means that they arrived 180° completely out of phase and interfered destructively canceling each other out the first bright fringe from the central fringe is a result of the path difference being the same as the wavelength lambda that means that these arrive in phase 2 interfering constructively resulting in a bright fringe the next dark fringe was a result of the path difference between being 1 and a half lambda and so on young's double slit equation is this w= lambda d / s where w is the fringe spacing that's the distance between the center of two bright fringes or two dark fringes for example d is the slit to screen distance and S is the slit separation note that that isn't slit width changing the actual width of the slits doesn't change the fringe width at all however this equation is an approximation and it can only be used when the screen distance is much much larger than the slit separation that's why the equation is likely to break down when we use sound instead if you have two speakers producing the same waves in phase if you walk at right angles to them you'll hear the volume fluctuate as you go through maxima and minima we can represent the pattern by drawing a graph of intensity against distance from the central max the intensity falls away gradually and they all have the same spacing but light can also defract when you just use a single slit as defraction occurs at both edges of the slit it gives this defraction pattern instead the two differences being that the intensity falls away quicker and the central max is double the width of the subsequent fringes don't forget that if you do this in reality it's always more accurate to measure 10 fringes than divide by 10 to get W when we do the experiment we're probably going to use a laser as it provides coherent light that's also monochromatic just one wavelength is emitted resulting in clearly defined fringes however using a candle that emits a spectrum of wavelengths Young saw the white light splitting into the different colors as red light has the longest wavelength it defracts the most so it could be seen on the outside edge of the fringes blue light has the shortest wavelength so it defracts the least so it could be seen on the inside edge of the fringes if you replace Young's double slit with a grating of very small lines separated by line spacing little D you only get constructive interference at a few points very far away from the central max we call these orders instead of fringes the central max is the zeroth order as we're dealing with large angles we can't use Young's double slit equations so we use the more accurate equation n lambda equals d sin theta often you'll be given the line or graating spacing in lines per millimeter which you need to turn into meters essentially meters per line is the reciprocal before it can go into the equation you can also be asked to find the maximum visible order in this case we want to find out what order is made at 90° sin 90= 1 so n= d over lambda let's say this ends up being 3.8 there's no such order so only orders up to three are visible also they might try to catch you out by asking hey how many orders will be visible rather than what is the highest order in this case you have to count the orders on both sides including the zeroth order so that would be seven in this case very cheeky just be careful with some multiple choice questions involving this equation you might get asked what will happen to the orders if the wavelength is half say in this case you don't look at theta at all as it's not proportional to anything in the equation instead we say n is inversely proportional to lambda if lambda halves n doubles that means that what was the first order is now the second order and a new first order is created between it and the zeroth order if d is changed we say n is proportional to d all particles are put into the groups hadrons and leptons electrons are fundamental particles and they include the electron muon that's just a heavy electron basically and neutrino that has no charge these all have a lepton number of one whereas their antiparticle equivalents have a lepton number of minus1 neutrinos can also be specifically electron neutrinos or muon neutrinos so you have to treat their lepton numbers separately if an interaction involves both electrons and muons hadrons are split into barons made of three quarks and mison made of two quarks a quark anti-quark pair so hadrons aren't fundamental particles the three flavors of quark we deal with are up down and strange up has a charge of plus 2/3 down and strange minus a third of course that's in terms of E only strange quarks have strangeness minus one for a strange quark plus one for an anti-range after all they are strange aren't they they all have a barrier number of plus a third so barons have a barrier number that isn't zero it can be one or minus one if you have anti-quarks in there too neutrons are up down protons are up up down so you can say the nud and the poo to help you remember them anti up anti-down and anti-range have the opposite charge and baron number pons are misons that don't have strangeness whereas kons are mison that do have it we can call them pi plus pi 0 pi minus and k + k minus k 0 etc to distinguish between them the electromagnetic force can affect any charged particle the exchange particle is the photon we might say virtual photon gravity's exchange particle is called the graviton but you know about those two already let's go with the two that are new to Alevel the weak force or weak nuclear force can affect any particle its exchange particle is the W plus W minus or Z0 Bzon the strong nuclear force only affects hadrons its exchange particle is the pion or gluon if pion isn't an option in a question this is what holds nuclei together the electrostatic repulsion between protons pushes outwards while the strong force pulls inwards okay a bit of gravity too but that's not much when these forces are balanced a nucleus is stable the range of the strong force is 3 to 4 fmpton but it switches from attractive to repulsive at 0.5 fame to stop the nucleus from imploding in any interaction charge baron number and leptton numbers must be conserved which is why there must be an anti-electron neutrino added in a beta minus decay equation to balance the leptton number we can draw fineman diagrams to represent interactions this will always be a weak interaction involving a wus bzon say for beta minus decay or w plus for beta plus decay or electron capture for normal beta minus decay we can also just say that one of the down quarks in the neutron is decaying to an up quark and that turns the whole thing into a proton it goes from nut to poo finally strangess rules any interaction involving lepttons must be a weak interaction regardless of stranges if an interaction only involves hadrons and strangeness is conserved it must be a strong interaction these can create strange particles though so say zero stranges going to +1 and minus1 for two different particles if strangeness isn't conserved in an interaction it must be weak not strong this happens when strange particles decay for example a k0 mison decaying into pi plus and pi minus mison in essence the weak interaction can destroy stranges but the total stranges can only change by one specific charge is merely the charge to mass ratio for a particle charge in kulum divided by mass in kilograms so the unit is kulum per kilogram so you'll always end up with a huge number for example for an electron we have the charge of 1.6 * 10 - 19 kum divided by its mass of 9.1 * 10 - 31 kg yes okay technically minus for the negative charge but it's really only the magnitude that we're concerned with a nucleus say helium 4 that's 2 * 1.6 * 10 - 19 for the two protons divided by 4 * 1.6 67 * 10 -27 for the mass of the four nucleons in the nucleus a singly charged ion well its overall charge is just 1.6 * 10 -19 we divide that by the total mass of the nucleus we don't really need to add the negligible mass of the electrons in this case the term radiation means any particle or wave that's emitted by something the electromagnetic spectrum is all radiation but they're all emitted by electrons all apart from gamma radiation that is gamma radiation is actually emitted by the nucleus of an atom if it has excess energy it's getting rid of gamma rays are high energy EM waves they can be dangerous as they can ionize atoms if absorbed by them knocking electrons off this can cause damage to the cells in your body and also cause cancer but there are two other types of radiation nuclei can emit too but these are actual particles and they're emitted when nuclei decay change isotopes with more neutrons are generally more unstable and likely to decay heavier nuclei like amarissium 241 decay by what we call alpha decay to become more stable the nucleus will emit a bundle of two protons and two neutrons what we can just call an alpha particle this is alpha radiation this is what the nuclear decay equation would look like for this to show that the nucleus has decayed into two parts the alpha particle which must have an atomic number of two and a mass of four and the daughter nucleus that's just the nucleus left over which of course is no longer going to be amorisium as it's lost protons to go from an atomic number of 95 to 93 turns out that's Neptunium but you'll never have to remember these you just need to worry about the numbers it's just maths 95 goes to 93 + 2 and the mass is similar 2 4 1 goes to 2 37 and 4 there is actually a nucleus that has the numbers 2 and 4 it's a helium nucleus lighter isotopes lighter nuclei like carbon 14 decay by beta decay or beta decay instead what happens is that a neutron in the nucleus turns into a proton and an electron but the fastmoving electron that's ejected by the nucleus escapes and we now call this beta radiation the mass of an electron is basically zero so we put that on top it has the opposite charge to a proton so we say it has an atomic number of minus1 now be careful here 6 goes to what + -1 no it's not five it's seven 6 is equal to 7 + -1 like we said a neutron has turned into a proton so the nucleus has gained a proton it's gone from 6 to 7 the mass however is unchanged so it's still 14 and don't forget to add the anti-electron neutrino on the end to balance the lepton number if a particle and its corresponding antiparticle meet say an electron and posetron they can annihilate and all of their mass is converted into energy in the form of two photons of EM radiation it's two photons by the way as momentum must be conserved and even though photons don't have mass they still have momentum weirdly enough if we're asked what the minimum energy or frequency of the photons made is we infer that they had no kinetic energy so we just say that energy equals 2 lots of mc^ squ we call this either rest energy or mass energy where c is the speed of light we can then say this is equal to two lots of hf which is the energy of a photon plank's constant times the frequency the shortcut of course is just to split the whole problem in half we can just say that mc^ squ for one of the particles equals hf for one of the photons the opposite can happen when a photon spontaneously converts into two particles if it has enough energy this is called pair production this time HF is equal to two MC squares because it's just one photon making two particles if the photon has more energy than the minimum energy needed the extra energy is supplied to the particles as kinetic energy it's the opposite for annihilation if they have lots of kinetic energy before that goes into the energy of the photons too electrons in an atom orbit the nucleus at specific or discrete energy levels n1 being the lowest energy level or ground state we can call it then n2 n3 etc an electron can be excited or raised to a higher energy level either by another free electron colliding with it giving it some of its energy or by it absorbing a photon however in this case the energy of the photon must match the difference in energy levels exactly otherwise it will just pass through either way after it's been excited the electron will pretty much immediately fall back down to the ground state emitting photons in the process we call this deexitation but it can take different routes down from N3 for example it can either go straight down to N1 or it can go to two first then down to one this means that there are three possible wavelengths or frequencies of photons that could be emitted when it deexites from N4 there are six different possible wavelengths the energy of each photon is equal to HF so the bigger the drop the higher the frequency of photon emitted and the shorter the wavelength is of course C equals F lambda if the incoming free electron or photon has enough energy however the electron can reach the ionization level which then means it is able to leave the atom leaving behind a positively charged ion the energy levels can be given in jewels or electron volts 1 electron volt is equal to 1.6 * 10 -19 jewels which should look familiar as the number is the same as the charge of an electron to convert jewels to electron volts and vice versa just think do I want a smaller number or bigger number you always want a very small number of jewels and a normalish number of electron volts after all that's why the unit is so handy sometimes mega electron volts are even better you can convert to electron volts and then you use the conversion factor of a million 10 ^ of 6 but I guarantee it's more helpful to remember that to go from jewels to mega electron volts the conversion factor is 1.6 * 10 -13 instead of -9 an emission spectrum is just a diagram showing the various wavelengths of photons that are being emitted by an object for example a star as different atoms or molecules have specific energy levels we can tell what particles are found in that star and we can also see how redshifted these wavelengths are too which we then use to determine the recessional speed of galaxies an absorption spectrum is obtained when we emit all wavelengths through a gas or plasma that's just an ionized gas and detect what wavelengths are not transmitted they don't pass through because they've been absorbed represented by black lines on the spectrum fluorescent tube lights strip lights consist of a cathode and anode that cause electrons to be accelerated through the mercury gas inside an electron will collide with a mercury atoms electron exciting it to a higher energy level when it deexites it emits a UV photon that's not useful for our eyes so it needs to be absorbed by the fluorescent coating on the inside surface of the tube it raises one of the electrons to a higher energy level in the coating and then that deexites via multiple levels emitting lower frequency visible photons instead we say photons have wave particle duality the photoelectric effect is the main piece of evidence that light acts like a particle shine light on certain metals and electrons will be liberated from the surface they pop off as they have absorbed the energy as kinetic energy how can we tell well if we have the metal in an evacuated tube in a circuit with an ammeter a tiny current will start to flow as the electrons are crossing the gap add a variable PD into the circuit and we can oppose this current until it reaches zero we call this potential needed the stopping potential VS we therefore say that at this point the kinetic energy an electron would have is equal to Q * V or E * V bear in mind that this technically gives the maximum kinetic energy an electron has after liberation as most electrons will have less energy due to them not being right on the top of the surface of the metal we vary the frequency of this incident light and plot the EKAX against this and we end up with a straight line the gradient is where plank's constant comes from it doesn't go through the origin though which means that the electron is not gaining as much K as the photon's energy going in it's lost some in the process of being liberated extrapolate this line back and we get this energy which we call the work function this is the minimum energy required for an electron to be liberated from the surface of a metal think of it as an energy toll the electron must pay to escape the minimum frequency needed to liberate an electron we call the threshold frequency f0 that means the minimum energy of a photon needed to liberate an electron is h * f0 and this is equal to the work function so here's the full equation ek max equals HF minus FI the kinetic energy the electron has left over equals the photon energy going in minus the work function the energy toll this proves the particle theory of light because if the frequency of light isn't high enough electrons won't be liberated no matter how intense or bright the light this shows that one photon is absorbed by one electron we say it's a onetoone interaction and that requires photons to be discrete packets of energy so light acts like a particle but any particle can also act like a wave if we fire electrons at a graphite target in an evacuated tube we see rings forming on the phosphorescent screen behind the only way that this would be possible is if the electrons are defracting as they pass by the graphite atoms and producing an interference pattern with maxima bright rings and minima dark rings electron defraction is the evidence that particles also have wave nature more on defraction in waves of course the wavelength of a particle is given by the debrogley or dero wavelength equation lambda equals h plank's constant over mv or p the momentum of the particle faster speed means smaller wavelength which means less defraction according to n lambda equals d sin theta from waves here's the standard graph showing how intensity varies with distance from the center note that it doesn't decrease to zero unlike defraction patterns for light at this point it's worth bringing up the fact that you might have to convert kinetic energy into momentum or vice versa the trick is this we know kinetic energy is equal to half mv^ 2 multiply both sides by m which gives me m=/ p that's momentum squared and let me just rearrange for p this comes in handy especially in multiple choice questions a current flowing through a wire will produce its own magnetic field the motor effect is when such a wire is in another magnetic field and it will experience a force the equation is F bill where F is force I is current in amps L is length of the wire in the magnetic field and B is the magnetic flux density essentially the magnetic field strength this is measured in Tesla rearranging the equation for B also shows us that the alternative unit for magnetic flux density is newtons per amp per meter note that this equation only works as it is if the current and magnetic field lines are perpendicular to each other but maybe it is worth remembering that if the wire is parallel to the field lines it will experience no force to find out the direction of the force however on the wire we use Fleming's left hand rule your thumb is force first finger is field middle finger is current make a janky gun with them where they're all perpendicular and bam freeze FBI just twist your wrist to line up your fingers with the current and the field always north pole to south pole and the way that your thumb is pointing is the direction of the force on the wire in this case upwards to measure the size of the force in reality we can put the magnet on a balance due to Newton's third law the magnet will also be pushed down with the same force calculate the force from the fake mass measured use an amter to get the current and a ruler to measure the length of the wire and boom you can calculate the magnetic flux density between the poles of your magnet if a free charged particle finds itself traveling perpendicular to a magnetic field it will experience a force equal to BQV for an electron or proton we can say that's FBV we still use Fleming's left-h hand rule with the current finger now being used for velocity however as it's used for conventional current in circuits which is technically the flow of positive charge we must flip our hand if it's a negative charge like an electron so our middle finger is pointing in the opposite direction to its velocity if it's a proton or positive ion though it's all good because we're thinking in three dimensions you usually see the circular path drawn on the page then the field going in or out of the page that's represented by crosses or dots as the force will always be perpendicular to the velocity in this case the particle will undergo circular motion as it's circular motion we can say that BQV equ= MV ^2 / R or M omega^ 2 R depending on whether we're dealing with actual speed or frequency in time period instead rearranging the first version we see that R is equal to MV over BQ so the radius of the circle is proportional to the mass and speed of the particle and inversely proportional to the flux density and particles charge with the angular version we know that V= omega R and so we see that the frequency of a particle moving in a circle is independent of its radius a cyclron is a machine that uses FBV to produce a stream of high energy particles like protons used for medical therapy purposes we have a source of these particles sitting in the center of two hollow metal D's so-called because of their shape a magnetic field is applied perpendicular to the D's which causes the protons to move in circles but they're not much use if they stay there so a PD is applied to the D's causing the protons to be attracted to and accelerated towards the negative D and away from the positive one that increases their kinetic energy and speed and therefore radius too half a cycle later though they must be accelerated across the gap again in the other direction so the PD must be reversed by the time they do this in order for that to happen we know that BQV or BEV equals M omega^ 2 R and replacing V with omega R and omega with 2 pi F we find that R disappears and F= BQ over 2 pi M that means it doesn't matter how big the radius of the circle is of a proton all the protons will have the same frequency and time period as they move in everinccreasing circles in the D's eventually they spiral out and that's how we get our beam whatever the frequency of the particles is that's also what the frequency must be of the alternating PD of the D's causing the particles to be accelerated people will often incorrectly say that the frequency must somehow be twice that of the particles frequency as it needs to flip twice every time they do a circle but flipping twice is exactly what a whole cycle is every cycle the particles cross the gap twice so that PD must be flipped twice too we can determine what particles we have in a substance by putting them through a mass spectrometer as BQV equ= MV^2 over R we see that R= MV over BQ if V B and Q are the same for all particles then the radius of circular motion is proportional to mass of the particle so that means particles traveling together initially will separate out according to their masses if they enter a magnetic field b is the same if they're all in the same magnetic field with Q is the same if they're all singly ionized that is they've all lost one electron getting them all to travel with the same speed is more difficult but it must be done in order for R and N to be proportional to do this we have a velocity selector before the main magnetic field it's a device with a small hole in the end we have a different magnetic field to the big one applied so the particles experience a force in one direction say upwards and an electric field applied in the other say downwards only particles for which these two forces are equal will carry on going straight through the hole for them EQ equals BQV electric force equals magnetic force the Q's cancel and we find that V equals E over B electric field strength over magnetic flux density those traveling too fast will be affected more by B and deflect and don't go through the hole too slow on the other hand and the opposite is true and they'll deflect the other way instead those going just the right speed for our two fields make it through now we have a beam of particles that are all the same charge and traveling in the same direction with the same speed there are two main designs number one the particles go through a semicircle in the magnetic field and hit a piece of film the further along the film that produce spots are the bigger the mass we can measure this diameter to get the radius and then calculate the mass from that brighter spots mean there are more of these particles hitting that spot so it's semi-quantitative the Posher version though employs an array of detectors placed at different positions which one a particle hits depends on its mass of course these can measure individual particles so give accurate data on relative abundance of particles in a substance flux is what we call the amount of magnetism that an area is exposed to symbol fi unit vebas but I also like to call them wubs flux density is equal to flux over area so another alternative unit for flux density we already have Tesla and newtons per meter but it's also vas per meter squared it's how concentrated the flux is there's a loop of wire in a magnetic field we can use the area of the loop with the known flux density of the field it's in to calculate flux fi= b * a often it will be a coil of wire that's capturing that flux to calculate the total flux affecting the coil we just take our flux and multiply by the number of turns in the coil n this is then technically called flux linkage but it works exactly like flux in calculations we can also therefore represent flux linkage as ban b * a * n if a loop of wire or part of a loop of wire experiences a changing magnetic field an emf is induced in it faraday's law is this emf induced is proportional to the rate of change of flux linkage now in reality the emf is equal to minus the rate of change of flux so volts are actually equal to v-bas per second but because we had that minus we technically have to say proportional instead of equal you have to know that definition but to all intents and purposes emf volt equals rate of change of flux the classic example is a wire being moved through a field perpendicular to the field similar to F bill if the wire was parallel to the field this wouldn't work the wire must cut the flux we might say we know the emf equals delta 5 over delta t but I'm going to drop the deltas and minus here for convenience also know n as it's just one bit of wire we know that flux is equal to ba this area is the area of flux swept through we can say in time t this area is equal to length time distance moved so emf equals BLD / T distance over time is speed so it turns out in this case EMF equals B LV nifty so why is that minus there well it's to show you that lens's law is also true the direction of EMF induced is such that it will oppose the change that caused it in other words an induced EMF or current will try to stop itself being made it does this by producing its own opposing magnetic field after all you know that's what a current flowing in a wire does this opposing force is the reason why you can't just turn a generator freely you have to put in work to overcome it that's equal to F bill fleming's right-hand rule tells us the direction of current induced same fingers with a thumb being which way the wire is being pushed another example of Lens's law is a magnet falling through a copper pipe the pipe experiences a changing magnetic field and an eddi current is induced in it that just means little circles of current this produces a magnetic field of its own that opposes the motion of the magnet that's why the magnet falls slowly through it at a constant speed if a loop of wire enters a magnetic field an emf will be induced as it enters and leaves while it's inside the emf is zero there are two ways of looking at this one way is saying that the flux filling up the loop or coil is changing as it enters and leaves but the flux stays constant while it's inside even though it's moving so therefore no emf the legit explanation though is that there is an emf being induced on both sides of the coil in the same physical direction while it's inside so they cancel each other out when entering or leaving there's only one constant EMF being induced on one side and so therefore we have an overall EMF we can again calculate this using BLV even if you have stationary coils you can still induce an EMF in one of them by applying AC to the other alternating current the AC produces a constantly changing magnetic field which the other experiences this wouldn't work with DC as that would only make a static magnetic field in this coil case emf equals a * delta b over t because a isn't changing in this case you'll likely be given the maximum flux density and you can just say that that's equal to delta b because it will fall to zero during the cycle and then we just use the time it took as well this of course is how transformers work generators use induction to produce an emf as the coil spins the flux through it goes from a maximum when it's perpendicular to the field lines to zero when it's parallel to the field lines the rate of change of flux is greatest when the flux is zero at this point we can just say that emf equals two lots of BLV much like our wire example the maximum flux is ban of course so if we want the maximum rate of change of flux we multiply by omega the peak emf induced in a generator is equal to ban omega the emf the rest of the time then changes sinusidally so the emf at any point is equal to ban omega that's the maximum emf time sin omega t or cos omega t depending on whether it starts at zero or a maximum respectively don't forget that whenever you use s or cos for spinny things they have a pi in the sign or cause you must have your calculator in radian mode in a power station instead of a coil spinning between magnets they have an electromagnet spinning between three pairs of coils called status these produce three separate currents that go to the national grid that's why you have more than two wires running along pylons incidentally overhead cables aren't made out of copper as that would be too expensive and they'd be too heavy we use aluminium it's light and it's a pretty good conductor but it's very ductile so to strengthen it we have a steel core running through the middle transformers are used in the national grid to change the voltage at which the electricity is transmitted through the overhead cables the cores are wrapped around a soft iron core get this into your head right now though there is or should be no electricity or current in the core the alternating current in the primary coil produces its own magnetic field and the iron core acts like a guide for it we use iron by the way as it's easily magnetized and demagnetized it works well as a guide this magnetic field then induces a voltage and current in the secondary coil in order for a current to be induced though a wire must experience a change in the magnetic field which is why we must use AC the ratio of turns in the coils is equal to the ratio of the voltages if the secondary coil has double the turns it has double the voltage and therefore half the current a step up transformer increases the voltage before it enters the grid this then reduces the current so less energy is lost due to heating the reason one goes up while the other one goes down is because electrical power is equal to voltage or PD* current V * I the magnetic field reason you have a different voltage induced in the secondary coil is because there are more or fewer turns in the coil to capture the flux that's in the core in an ideal world the power in and out of a transformer should be the same that would mean that it's 100% efficient so V and I are inversely proportional we can therefore say that V * I for the primary coil is equal to V * I for the secondary coil if a transformer isn't 100% efficient it's always the current that takes the hit in this case you stick the efficiency in front of V1 I1 for example 0.8 if it's 80% efficient and then you're likely to use that with V1 over V2 equals N1 over N2 we know the lenses law explains why it's hard to rotate a coil in a generator the induced current is fighting back with its own opposing magnetic force due to the motor effect you can't have one without the other balenc's law also applies to motors too when a motor spins the generator effect is also in play when the motor is turned an EMF is induced in a coil that opposes the EMF driving the motor that might sound like a bad thing but it's not if there's six volts supplying to a motor but the opposing what we call back EMF is 2 volts in the opposite direction that means we have four volts across the motor instead of six and therefore reducing the power loss due to heating too because P= I^ 2 R if a motor is under heavy load it spins slowly the back EMF is small and the current is high too heavy a load and the motor can burn out i'm sure you've experienced this when you smelt the smoke coming from your scale electric car when you're holding it down but still pushing the throttle if a motor is under light load though it spins faster the back emf is large and the current in the motor is small great a Z and N are the symbols used for mass number atomic number and number of neutrons respectively here's a graph of N against zed as you can see isotopes generally sit above the N= Z line which means they have more neutrons than protons the further they are from this curve the more unstable they are and the more likely they are to decay to get closer to the curve lighter isotopes are generally beta minus emitters heavier isotopes are alpha emitters beta plus emitters don't exist naturally as they require proton doping to produce the right isotopes which puts them under the curve we can zoom in on this graph to show the decay chain an unstable isotope will move through they can also take different routes to end up at the same stable isotope they're always diagonal lines in this case but just be aware that you could also get a graph of a against zed instead in this case you get a horizontal line for bic as the mass doesn't change briefly metastable states gamma emitters aren't drawn on these graphs because of course the nucleus doesn't change what usually happens is that an alpha or beta decay occurs leaving the nucleus at an excited energy state much like an electron it will usually deexite instantly emitting a gamma photon as well this means it's hard to get a purely gamma emitting source however some isotopes can decay and then remain at the excited state for a long time after that before releasing that photon this is called a metastable state the example we usually use is malibdinum 99 this decays via beta minus leaving technesium 99 in a metastable state this is then filtered from the undeayed malibdinum to then be used as a pure gamma emitter for uses such as medical traces or radiotherapy in reality the relative mass of nuclei is based on the relative atomic mass unit u this is defined as 112th the mass of a carbon 12 atom and we find that neither a proton nor a neutron have a mass of exactly one U even if a helium nucleus had a mass of exactly 4 U if you manage to split it up into its constituent nucleons two protons two neutrons it will end up having a mass of more than 4 U how's that it's because it requires work to do that to separate them out to overcome the strong nuclear force this work is turned into mass and we use E= delta MC^² for that delta M being what we call the mass defect what's the difference in mass between before and after of course E= MC^² involves jewels in kilograms which means you'd have to convert the mass from U first but here's the shortcut one U of mass converted into energy gives you 931.5 mega electron volts this will be in your formula sheet it's a much nicer conversion factor to work with if we draw a graph of average binding energy per nucleon against mass number we end up with this shaped graph now to all intents and purposes average binding energy and total binding energy of all particles involved follow basically the same patterns so they're essentially the same thing so maybe that will help you understand what's going on here remember that binding energy is the energy needed to separate a nucleus which means that the one at the top requires the most energy to separate out the nucleus this is the most stable isotope therefore and it happens to be iron 56 that's why an iron core can be left at the center of a star that's used up all of its fuel the little spike on the left is helium 4 by the way it's an uncharacteristically stable isotope compared to its immediate neighbors that's why it's emitted during alpha decay however when nuclear events occur such as fision fusion or decay they don't involve nuclei being completely split into their constituents or vice versa but rather just turning into other nuclei which are more stable meaning they always move up the curve as their total binding energy and therefore their average binding energy per nucleon 2 has increased we can think of them as falling down a potential well meaning they would need more energy to get out fision is the splitting of a nucleus whereas fusion is the joining together of them in both cases energy can be released so long as the binding energy has increased they fall down this imaginary potential well and therefore climb this curve that means the isotopes to the left of iron 56 release energy if they undergo fusion to the right of it the same is true if they undergo fision instead if you're asked to calculate the binding energy in an interaction all you have to do is find out the mass defect in use and then times by 931.5 to give you the binding energy in mega electron volts fision usually needs to be induced for example a uranium 235 nucleus in a nuclear reactor's fuel rod will absorb a neutron turning it into uranium 236 which is more unstable and it splits roughly in half producing two or three more neutrons that go off and do the same in the process energy is released most importantly as kinetic energy which heats up the coolant often CO2 which then supplies heat to turn water into steam to turn a turbine etc hence why such a reactor is often called a thermonuclear reactor the neutrons released from fision are actually traveling too fast to be absorbed by the uranium nuclei so they must first be slowed down by a moderator usually water or graphite to what we say thermal speeds a reactor doesn't have an off switch though so control rods are used to increase or decrease the rate of fision often made from boron neutrons will be absorbed by them without causing further fision they're lowered or raised between the fuel rods depending on the output needed don't get confused with the moderator it's not there to slow down the reaction in fact it's the opposite it speeds up the rate of fision by slowing down the neutrons so they can be absorbed and cause fision nuclear waste is awful stuff it's hot and very radioactive so it must be disposed of safely it's left to cool in ponds first and then it's vitrified in glass which is non-porous and stored deep underground away from water sources or fault lines so there's a near zero chance that it can affect living things activity is given by the simple equation a= lambda n where n is the number of undeayed nuclei not neutrons in this context and lambda is the decay constant in seconds to the minus1 but it could also be another unit for time to the minus1 instead the decay constant is the probability that a single nucleus will decay in that amount of time i've dropped the minus that's sometimes in the equation as that's only important when deriving the exponential decay equation which you don't need to do for your exam so we're not doing it now but the equation is this a / a 0 that's just activity at time t compared to initial activity is equal to e to the power of minus lambda t where t is the time and the unit of that must be the equivalent of the decay constant in order to work you might see the equations written differently in your formula sheet but I think it's useful this way for us to see that it's merely a ratio of activity now to initial activity so it's just a number between 0 and one it also works for n / n0 that's number of nuclei left and mass 2 if we need to rearrange for lambda or t we do the natural log on the other side you might say l if you're weird so natural log of a over a 0 equals minus lambda t the minus is important here if activity is halved the time t is the half-life by definition so that's where t half equals natural log 2 over lambda comes from if you're given a graph of activity you'll invariably have to find the half-life from it and then calculate lambda as natural log 2 over this half-life then reuse lambda in the full equation as gamma radiation is emitted in all directions it falls away inverse squarely with distance so intensity is inversely proportional to distance squared so we can say that I1D1^ squar equals I2D2^ squ intensity can also be replaced with count or count rate the amount of radiation detected over a certain time say a minute don't forget your background count of course if you draw a graph of count against 1 / d^ 2 you'll likely get a straight line that doesn't go through the origin this is because the distance measured won't usually take into account that the radiation isn't being absorbed right at the end of the tube so there might be a systematic error of a couple of centimeters or so if the graph curves when you get close to the source then that implies that there's probably some alpha or beta being emitted as well you should remember from GCSE that it was Ernest Rutherford who discovered that the nucleus was positive and small relative to the size of an atom but he also estimated how much smaller it was too by comparing the number of alpha particles deflected back through an angle greater than 90° to those that went through this proportion gives you the ratio of the area of the target circle of the nucleus compared to the atoms the root of this gives you the ratio of the diameters however a more accurate way of measuring the diameter of a nucleus is with electron defraction as you know from quantum electrons will create a defraction pattern when incident on atoms or nuclei the angle between the central max and the first order can be used in our defraction equation d sin theta= n lambda to find the diameter lambda being the debrogley wavelength if you plot a graph of any log of the radius against the log of mass number a also the number of nucleons in the nucleus we get a straight line with a gradient of a third and a y intercept we can say is equal to log r0 combining the right side and dlogging we end up with r= r0 * a ^ of a3 often you'll be asked to compare two nuclei to find an unknown radius or nucleon number so as r is proportional to a the 3r we can say that r1 / a1 the 3r is equal to r2 2 over A2 to the 3 scanning a body is useful as it's noninvasive we don't have to open somebody up to see what's wrong in a lot of cases we need to make X-rays for an X-ray scan obviously we do this by having a cathode and anode with a high PD applied the cathode is heated and electrons pop off that and are attracted to the anode which includes a heavy metal target like tungsten the electrons collide with the tungsten exciting its electrons and they emit X-ray photons when they deexite these photons need to be attenuated by what you're scanning if you want to produce an image on a film or sensor to attenuate radiation means to reduce the intensity of it there are four ways that this happens simple scattering also known as rally or Thompson scattering when a photon incident on an atom doesn't have enough energy to ionize it but interacts with it in such a way that it changes direction without a change in energy the photoelectric effect you already know this one if a photon has enough energy it will be absorbed by an electron on an atom liberating it the Compton effect is kind of a mix of the first two the photon causes an electron to be liberated we can say it's scattered itself but the photon isn't completely absorbed instead it is scattered too with a lower energy than it had initially finally pair production again you know this one if a photon merely comes close enough to an atom to interact with it the photon can spontaneously turn into two particles a particle and its antiparticle usually an electron posetron pair whatever happens the intensity of the X-rays is reduced a different amount depending on what is doing the attenuating bone soft tissue etc we have an equation to describe the exponential decay of the intensity but it's not with time it's with distance through the material i over I0 intensity over initial intensity is equal to E to the minus Mux where X is the distance it travels through the material and Mu is the attenuation or absorption coefficient just a constant that's different for every material and as you know that this power on the E can't have a unit mu must have the unit meters to the minus one bone has a higher mu than soft tissue which means that this ratio is less for it so less is transmitted to the film which is why bone show up on the scan And there is that contrast between bone and soft tissue but what if you want to scan a part of your body with no bone say your bladder or digestive system for that a liquid or medium with a high mu is injected or ingested it might include barium or iodine that provides the contrast needed that's why we call it exactly that a CAT scan computerized axial tomography sometimes just shortened to CT scan is an X-ray on steroids the X-ray tube is rotated around the patient with a ring of detectors sensing how much they're attenuated at every angle this data is then processed by a computer to build up a 3D image of your internals including soft tissue contrast isn't needed necessarily but it is often used to speed up the scan it tells you so much more than a normal X-ray does but you do get a huge dose of radiation which means it's unusual for you to have more than one a year say a tracer is a gamma emmitting liquid that is injected into a patient something like florine 18 or technesium 99 which you know about from nuclear the gamma rays are detected by a gamma camera generally a parallel hole design only those traveling parallel are detected and that allows an image to be formed compensating for the fact that lenses don't really work with gamma rays so can't be used to focus them the photons that make it through the columator reach the scintillator that's a material that absorbs gamma photons and emits thousands of lower energy visible light photons per gamma photon absorbed finally these visible light photons reach the photo multiplier tubes PMT for short which convert the photon energy into an electrical pulse which is then used by a computer to produce an image this allows doctors to see where and how blood is flowing and if there are any blockages or internal bleeding for example posetron emission tomography or PET scans use annihilation a posetron rich substance is injected into the patient the posetrons very quickly meet electrons and annihilate producing a pair of gamma photonss that travel in opposite directions to the ring of detectors around the patient and their point of origin is able to be calculated ultrasound is any sound that has a frequency higher than 20 kHz these can be used to scan a patient's soft tissue this is because much like light sound waves are partially reflected when they reach a boundary between two materials or mediums at least if they have different acoustic impedance the symbol we use for acoustic impedance is zed and it's just equal to the density of the material row times the wave speed C if two mediums have the same impedance we say they're acoustically matched and no reflection occurs this is why gel needs to be used between the transducer and the patient's skin to minimize reflection before the waves are inside the body the ratio of the intensity of reflected wave to incident wave is given by the difference in impedance is squared divided by their sum squared the transducer is what emits the ultrasound and detects the reflected waves it consists of an array of po electric crystals crystals that change shape slightly when you apply a PD to them applying an alternating PD makes them vibrate at that same frequency allowing for very high frequency sound waves to be formed not only that though when sound waves are incident on them a tiny PD is induced in them which is then detected in fact some normal microphones use the PO electric effect to produce a signal in the same way there are two main ways ultrasound scans are performed an A scan involves sending pulses downwards into the patient which are then reflected at the various boundaries at different depths the time taken for them to return is measured from the resulting echoed pulses detected and the depth of these boundaries are calculated building up a one-dimensional image of what's below doing lots of A scans while moving the transducer sideways is a B scan which of course allows us to build up a 2D image of the patient's internals ultrasound scans don't provide nearly as much detail as other scans but they are beneficial for a couple of reasons firstly they don't produce any harmful radiation which is why they are the scanner choice when we want to see what's going on with a baby in a mummy's tummy secondly they show layers of soft tissue whereas X-rays can't do that and CT scans generally need added contrast they're also cheap quick and easy ultrasound can also be used to measure blood flow rate in veins and arteries using a Doppler scan so-called because it measures the change in wavelength of the waves reflecting off red blood cells traveling towards or away from the transducer similar to red shift in astrophysics we take the change in frequency over original frequency and equate it to two lots of the speed of the blood cells over C the wave speed there's a two in there of course because the wave is going there and back unlike with galaxies just emitting the light to us in astro but as the transducer can't be directly in line with the blood vessel we multiply by cos theta to compensate for the fact that it's at an angle to it when it's on the skin you always want to make accurate measurements reducing any errors and uncertainties to a reasonable degree when measuring any length the ruler should be as close to the object as possible to avoid parallax error you want your eye to be in line with the point on the ruler too for the same reason parallax error is a random error by the way it won't be the same every time if it's not possible to have them close together say when measuring the extension of a spring you can use set squares just a right angle triangle even made out of a piece of card to accurately show where it lines up on the ruler it's also a good idea to use one between the ruler and a flat surface say the table to ensure that the ruler is indeed vertical the resolution of a ruler or a meter rule is usually 1 mm that's the smallest difference it can measure there can be large random error when using a stopwatch due to reaction time to mitigate this you should always do repeats and calculate a mean this is also true when taking measurements say with a venue caliper and micrometer too for example when you're finding the thickness of the wire in the young modulus and resistivity experiments digital instruments will give you a value on the screen all you have to do is record the readings say a digital voltmeter however sometimes it's more useful to use an analog instrument in this case your eye must be in line with the needle to avoid parallax error sometimes they have a mirror behind it to help you you might have to choose one that has an appropriate full scale deflection for your measurements that is the maximum measurement it can make this is usually 1 2 5 10 50 100 etc let's say that we figure out that the maximum voltage we're going to have to measure is 6 volts the appropriate full scale deflection would be 10 volt in this case 1 volt would be too small of course and 100 volts would result in inaccurate readings because the resolution would be too large and the uncertainty would be massive a micrometer has a much higher or I prefer saying smaller resolution usually 0.01 mm that's 1/100th of a millimeter first it's a good idea to calibrate it close the jaws with nothing between using the ratchet on the end this ensures you don't overtighten them which would result in the jaws being compressed giving the wrong measurement if it says 0.00 mm at this point it's all good if not there is a zero error a systematic error with any measurements you make after that systematic meaning it will be the same error every time to mitigate this just subtract the error from every measurement when measuring the thickness of an object with it say a wire we use a ratchet again until it stops see how many half millimeters the barrel has moved past so in this case it's 1.5 millime then add on the hundreds using the scale on the barrel itself which is 24 here so 1.5 + 0.24 mm gives you 1.74 mm if it read 0.01 mm when we were calibrating this measurement would actually be 1.73 mm in reality if it read minus0.01 01 mm on the other hand this would be 1.75 mm instead we've taken away a minus number which makes the measurements bigger veneer calipers work in a similar way the zero on the slide tells you how many millimeters then the veneer scale gives you the 10th or sometimes 20th that's 0.05 mm you have on top a veneer scale works because its lines are spaced such that they don't line up with the main scale however one of them will line up with the main scale that's what you take as your tenth so this would be 8.7 mm they can also measure the depth of a hollow object say a tube using the rod that comes out of the other end that goes in the tube and the edge of the caliper rests on the top then you just read it as normal uncertainty is the idea that even if you avoid parallax error zero error etc no measurement will ever be completely accurate because there's always a bit of rounding involved in every one you make for example using a thermometer with a 1° C resolution will require you to decide if the liquid is closer to say 27 or 28° in chemistry you'll likely say that this temperature therefore is 27 plus or minus 0.5° they say the uncertainty in any measurement is half the resolution now you will likely get the mark if you put that in a physics exam as well but many argue including myself that there's every possibility that you might think the liquid is closer to 27 when it's actually closer to 28 that means that the true value is not within your uncertainty that's why most questions expect you to state the uncertainty as being equal to the resolution itself so this would be 27 plus or minus one° C it's being a bit more honest and erring on the side of caution the uncertainty of a digital instrument can be inferred to be just one of the last digit so if you're given a voltage of 5.32 volt you can assume that the resolution used was 0.01 volt which will also be the uncertainty finding the uncertainty in a mean value just find the range of the values you add it up that's the maximum takeaway the minimum by the way and not the first takeaway the last which people often do and then you have it this gives you an absolute uncertainty which you can then turn into a percentage uncertainty if needed drawing a line of best fit on a graph allows you to find a gradient however sometimes the points will have error bars to reflect the uncertainty if so you can draw a line of worst fit too this is the line that has the maximum or minimum possible gradient it's your call that still crosses every bar note that that might mean that it doesn't hit the ends of the first and last error bar then find the difference between the gradients of it and the line of best fit to get the uncertainty alternatively you could just draw the two lines of worst fit and find the mean of the two to get the gradient and the uncertainty will be half their range if you add or subtract two values that have an uncertainty then the total uncertainty is the sum of the two uncertainties if you're using a ruler that you can move so the zero can be lined up with the edge of whatever you're measuring the uncertainty will be equal to the resolution however if the ruler can't move and you have to find the measurement by finding the difference between two points then the uncertainty will be double the resolution this is because we've essentially just combined two uncertainties or rather two absolute uncertainties absolute uncertainty is a normal uncertainty it has a unit just like the measurement but what if you're multiplying or dividing values that have uncertainties instead in this case you convert the absolute uncertainties into percentage uncertainties and you then add them all up regardless of what's being multiplied or divided so let's say we have these measurements which we've put into an equation to calculate an answer here are the uncertainties in percentage form as well and I've added them up to give the total percentage uncertainty uncertainties always increase by the way regardless of whether you're adding subtracting multiplying or dividing the only exception to this is when you're rooting a value so if you wanted the percentage uncertainty in the square root of V we just half the percentage uncertainty in V you get the idea once you have the total percentage uncertainty you need to turn this back into an absolute uncertainty for the final calculated value just multiply it by the percentage and certainty as a decimal and boom you're done we usually try to obtain a straight line graph in physics experiments which allows us to find a gradient and therefore the relationship between the two variables on the axis variables aren't always directly proportional though of course but if we know the relationship we can plot the right thing on the axis so we still get a straight line graph for example with the stationary wave on a string experiment we know that the frequency is proportional to the square root of the tension or f2 is proportional to t so if we plot f^2 on the y-axis t on the x-axis we'll get a straight line if we have some sort of inverse proportional relationship then one over something must go on one of the axes log graphs are powerful because they can allow us to find the relationship between two variables if we don't know it prime example of this is what Kepler did to coin his third law he saw that the further away planets were from the sun the longer their time period but he needed to find out exactly how they were related as t clearly wasn't proportional to r but r to the power of something instead if you take the planets in our solar system and plot log t against log r we get a straight line doesn't matter what log base we use by the way it works even for natural log ln a graph with log values always gives a straight line the gradient is the important thing here usually this is what gives us the relationship in this case the gradient is 1.5 or 3 over2 the general straight line formula is y= mx plus c or in this case log t= 3 log r plus some constant which is our y intercept let's call it log k from there we have two log identities we need to use first we can take the three halves and make it the power on the r then we can also say that the sum of two logs is equal to the log of the two things multiplied so on the right we have log kr to the power of 3 halves dlogging both sides we see that t is proportional to r ^ 3 or t ^2 is proportional to r cubed even if logs aren't involved you still need to be able to interpret a graph in terms of y= mx + c to produce an equation bear in mind that sometimes it's only one of the axes that will be a log just remember that if you dlog one side in this case it's 10 the power of the other side or e to the power of the other side if it's natural log finally it's possible you'll be given a graph that has an actual log scale it's nonlinear in this case the first line is one of that order and the lines then get smaller approaching nine and then it goes into the next power up so 10 20 30 up to 90 then 100 200 etc reading off these graphs often requires you to make a judgment call with values between the lines so just remember that going halfway to the next line isn't actually half the difference it's not five but it's more like three leave a like if you found this helpful and you can click on the card to go to the playlist for all the papers