In this video I'll be going through my notes on level 3 mechanical systems. A link to the PDF and a list of corrections is in the video description. Scalars and vectors. A vector is a quantity with both size and direction, a scalar only has size, no direction. Examples of scalars include pressure, temperature, time, mass and energy.
Now time is one that you could possibly argue, but at this level of study we assume it to be a scalar. Examples of vectors include displacement, velocity, acceleration, force and torque. Basic motion. Displacement is position change, we measure it in units of meters and the scalar version without direction is called distance.
And so if I was going to go from A to B and I went there and then I went back if the distance between them was 20 meters. The distance that I would have traveled would be 40 meters there and back, however with knowledge of direction we can recognize that overall my position hasn't changed since I'm back where I started and therefore my displacement would be 0. Velocity is rate of displacement change, we measure it in units of meters per second, which we can write as meters divided by seconds or equivalently ms minus 1. Our scalar version is called speed, and the equation we can use to describe it is that the velocity is equal to the displacement divided by the duration over which the displacement occurred. Acceleration is the rate of velocity change, we measure it in units of meters per second, per second, and we write it as ms minus 2. As an equation, we can write that the acceleration is equal to the velocity divided by the duration.
Equating vectors. Like scalars we can form equations with vectors. We can use the following techniques. First of all circles are zeros where vectors forming closed paths equal zero.
So if we have a vector A and then we add on a vector B, add on a vector C and then finally a vector D, we see that all of these vectors form a closed path and that that path leads us back to where we started. so that the sum of all these vectors is just zero, back to where we started. In our second example, we have our vector A, B, and C, which form a closed path and therefore also must equal zero. No matter which order you add them and how many vectors you use, a closed path of vectors is always going to equal zero. Equivalent paths.
A vector or set of vectors that start and finish in the same place are equal. So if we have our A... plus b and that we can also get to this same destination with just a single vector c our a plus b must equal as c furthermore if we have our a plus b and we can also reach that same destination with c plus d then a plus b must equal c plus d finally we have our a plus b plus c which puts us at the same destination as just d therefore a plus b plus c must also equal d. Trigonometry refresher. In this course we'll extensively be using trigonometry, so on this page we're going to briefly go over the basics.
Fortunately every triangle that we will be applying mathematics to in this course is going to end up to be a right angled triangle, which makes the mathematics a lot easier. Our longest side is our hypotenuse, and our other two sides depend on where the angle is. If we define the angle as being here, then our adjacent side is adjacent to that angle, and our opposite side is opposite the angle.
If I define the angle to instead be here, then our adjacent and our opposite would be the other way around. Pythagoras gives us the equation that the hypotenuse squared is equal to the opposite squared plus the adjacent squared. With a bit of rearranging we can solve this for each side.
SOH CAH TOA is a way of remembering the three trigonometric ratios. SOH is that the sine of the angle is equal to the opposite over hypotenuse. CAH is that the cosine of the angle is the adjacent over hypotenuse.
And TOA is that tan of the angle is equal to the opposite over adjacent. Vector components. It is often useful to separate a vector into horizontal and vertical components. Knowing the total magnitude and the angle, we can use Sohcahtoa. So for our vector where the magnitude is v and our angle to the horizontal is theta, we have our horizontal component and our vertical component and also a right angle between them.
We're able to use our Sokotoa because this is a right angled triangle. If we're going to use our Ka relationship, that tells us that cosine of the angle is equal to our adjacent which is our v horizontal divided by our hypotenuse which is v. multiplying both sides by v and swapping the sides around, we end up with this equation here. For our v vertical we can use the SOF relationship, where sine of our angle is equal to our opposite which is our v vertical, divided by our hypotenuse v. Once again multiplying both sides by v and swapping them around, we end up with this equation here. Momentum.
Whereas we have previously used velocity to describe the amount of motion an object has, this doesn't take into account the mass of the object. For this we use momentum, which has the units of kgm per second. As an equation, our momentum is equal to our mass times velocity.
We could determine the momentum of a bee with a mass of 0.01 kg and a velocity of 10 m per second to be 0.01 kgm per second. We could also consider the momentum of a car traveling at the same velocity, but with a much larger mass. Because of its larger mass, we have a much larger momentum.
Impulse. The longer a collision takes, the smaller the force. This phenomenon is called impulse.
As an equation, our force is equal to the change in momentum, divided by the duration over which it changed. Consider a force time graph. A large force over a short duration would look something like this, whereas on another graph, we could represent a small force over a long duration as something like this. The area underneath both graphs is roughly the same, therefore we have the same change in momentum.
If you consider a car moving with a particular momentum, we could imagine stopping it suddenly by slamming on the brakes, or gently applying the brakes and coming to a slow stop. Conservation of momentum. If there is no unbalanced external force, momentum is always conserved.
In other words, the total momentum before will equal the total momentum afterwards. For example before, if we consider a orange ball with a momentum of 20, and a green ball with a momentum of negative 30, negative because it's in the opposite direction, our 20 plus our negative 30 give us negative 10. After they've collided. We could imagine our orange ball with a momentum of negative 25 and our green with a momentum of 15. Negative 25 plus 15 gives us negative 10. A momentum before and after is the same, therefore momentum has been conserved. Here we see a man mixing mentos and coke to manufacture a rocket. The reaction expels gas and fluid out the bottom.
Via conservation of momentum, the downwards momentum that the fluid leaves with provides the bottle with an equal and opposite upwards momentum. As simple as this may be, the physics behind this is the same physics that gets us to the moon. 2D Momentum Example Ball A collides with stationary ball B.
They move off at right angles to each other. Determine the speed of ball B after the collision. And so before we have our ball A moving with a velocity of 1.80 mps with a mass of 1.70 kg, whereas ball B is stationary and with a slightly smaller mass. Afterwards, the balls move off at right angles to each other, ball A with a velocity of 1.10 mps and ball B with a velocity that we're trying to find. To determine the final ball B velocity, we need to know the final ball B momentum, which we can find using conservation of momentum, if we know the final ball A momentum, and what the initial momentum was, since our initial momentum is going to equal the final ball A momentum, plus the final ball B momentum.
So let's determine the initial momentum. Since ball A is the only object with momentum before, our momentum before is just its mass times velocity, giving us 0.306 kgm per second. Now to determine the final ball A momentum, momentum is our mass times velocity, we know its mass, and we know its velocity, which gives us 0.187 kgm per second. We can now use conservation of momentum to determine the final ball B momentum, knowing that our initial momentum of 0.306 must equal our final ball A momentum plus our final ball B which is what we're trying to find. Since we know that our final momentums are at right angles to each other, we end up with a right angled triangle.
We can therefore use Pythagoras to find our unknown side, which gives us 0.242 kgm per second. We now have all we need to determine the final ball B velocity. Since we know that momentum is mass times velocity, we can rearrange that to get that the velocity is momentum divided by mass.
Knowing our final ball B momentum and our ball B mass, we get 2.02 meters per second. Center of mass. The center of mass of a set of objects is the weighted average position of their masses. If two objects are balanced about a pivot, the point at which they balance is their center of mass. So if we have two identical Earths, the center of mass of two identical objects is halfway between them.
Whereas if we replace one of our Earths with a much larger Jupiter, as the mass of Jupiter is much larger than Earth's mass, the center of mass is much nearer to Jupiter. If we consider two masses, m1 and m2, we can determine the x coordinate of the center of mass between them, knowing the x coordinates of our masses. The equation is basically a mathematical version of this statement.
And so we take the positions of our objects and weight them by their masses, and then take the average. Now in many situations you can make your x1 equal 0, which makes x calm the distance from mass 1 to the centre of mass, and greatly simplifies the mathematics. I have here a glass, two forks, and a matchstick. Balancing these just right, and spending an embarrassing amount of time doing it, we see what initially appears to be counter-intuitive. All the weight is on the end of the matchstick, so why doesn't the matchstick topple?
If we consider where the match is pivoting and draw a line through it, what we see is that we have a roughly equal amount of mass on either side of our line, meaning that our match fork structure has its center of mass above this pivot, and since their combined force of gravity acts from this point, it has a distance of zero. As the distance to our pivot is zero, no torque is produced. Center of mass in 2D.
Finding the center of mass of objects in 2D is done by finding the XCOM and YCOM separately. So if we establish an XY coordinate system, and our mass 1 with an X coordinate and a Y coordinate, as well as our M2, We can imagine their center of mass with coordinates as well. In this case we just analyze our x and y situations separately.
Center of mass in collisions. The motion of the center of mass is unchanged before and after collisions, provided there are no external forces. And so if we imagine our mass 1 and our mass 2 having had collided, the motion of the center of mass between them provided there are no external forces, is going to maintain its original motion.
As you might guess from the language that we're using, this is really just a statement of conservation of momentum. This is a melon with no momentum. Moments after the melon explodes, let's ask the question what is the momentum now?
If the melon exploded in space with no forces to impede it, we could add up the momentums of every single piece and find that they add to zero. However, in this case, we have two key influences. We have a force from our table, applied upwards to all pieces in contact with it, and we have the force of gravity, applied to every single piece. From the perspective of our melon, these two forces are external, therefore the melons momentum is not conserved.
Conservation of energy. Energy is never created or destroyed, only transferred into different forms. We call this conservation of energy. We could imagine 10 joules of electrical energy being converted into 3 joules of light and 7 joules of heat. The total energy hasn't changed, only the forms that it's in.
We could also imagine 35 joules of gravitational potential energy being converted into 2 joules of air resistance and 33 joules of kinetic energy. Finally we could have 100 joules of chemical energy converted into 10 joules of light and 90 joules of heat. Work.
Work describes the energy used to perform some action. It is effectively another word for energy. It therefore has the same units of joules. The work performed when applying a force over a particular distance. So here we see a force being applied to a mass on a spring over a particular distance.
is given by the equation that the work is equal to the force times distance. Power. Power describes the rate at which energy is consumed. It is measured in joules per second, which are commonly called watts.
We can describe it with the equation that our power is equal to our energy divided by the duration over which it's consumed. Examples include a 25 watt stereo consuming 25 joules every second. 25 joules divided by 1 second is 25 joules per second, or in other words, 25 watts. A 10 watt fan consumes 10 joules every second, since 10 joules divided by 1 second is 10 joules per second, or in other words, 10 watts. Force.
A force is an influence that acts to accelerate objects. We measure it in newtons. The specific nature of forces is described by Newton's laws of motion. Newton's first law states that an object will only change its motion, or in other words its velocity, if a force acts upon it. Newton's second law states that when a force acts upon an object, it will accelerate proportional to its mass.
We can describe this with our equation force equals mass times acceleration. Newton's third law states that forces exist in equal and opposite pairs. You may also have heard the interpretation that every action has an equal and opposite reaction. I have here an RC helicopter, and three things I'd like you to try to explain. How does it go up and down, how does it go back and forwards, and how does it rotate?
Here's some footage of it flying to help you think. Now for the explanations. To move up and down, the helicopter spins its main rotor, which is shaped to apply a downwards force to the air. In return, the air applies an equal and opposite force to the rotors.
If this force is greater than the force of gravity, the net force will be upwards, and the helicopter will accelerate up. If the force is equal, there will be no net force and the helicopter will remain at the same altitude. If the force is less than the force of gravity, the net force will be downwards and the helicopter will accelerate downwards. To move forwards and back, the helicopter spins its back rotor. If the rotor forces air upwards, it will experience a downwards force, which will tilt the helicopter backwards, giving our main rotor force a backwards component, which will accelerate our helicopter backwards.
If the back rotor blows air downwards, it will experience a force upwards, tilting the helicopter forwards, giving the force on the main rotor a forwards component, and accelerating the helicopter forwards. The helicopter is able to rotate using its two main rotors. These spin in opposite directions, and if they spin at equal velocities, they will apply equal and opposite torques to the helicopter.
In this case the torques cancel out and there's no net torque on the helicopter, but if one of these rotors is slowed, the torques will no longer equal, and the helicopter will experience a net torque causing it to rotate. In this simulation, I've positioned balls of helium, glass, rubber and stone above the ground. Upon release, we see the glass, rubber and stone fall at different rates. This happens because the force from the air friction affects them in proportion to their different masses. Because our ball of helium is less dense than the air around it, the buoyancy force exceeds its force of gravity.
As a result, it accelerates upwards. Repeating the experiment but removing the air without the force of buoyancy and the force of air friction, all four materials accelerate downwards at the same rate. Circular motion.
We describe objects in circular motion using the following terms. So we have our object, our bowling ball, which is undergoing circular motion at a radius r, which is half the circle's width. We can also define our circumference which is 2 pi r which is the length of one lap. A revolution is a term for a complete lap, the period t is the seconds per revolution whereas the frequency is the revolutions per second. As I've tried to indicate with the language these two are reciprocals of each other so that frequency is 1 over period and period is 1 over frequency.
The velocity of our object can be considered to be distance over time as always, where the distance for one revolution is 2 pi r and the time for one revolution is just our period t. Centripetal acceleration. As an object changes direction, its velocity changes, even if the size of the velocity does not. So because velocity is a vector, changing the velocity can be done by either changing the magnitude or in this case changing the direction.
The object has therefore accelerated, we call this centripetal acceleration and the force causing it the centripetal force. This is not a force itself but rather a role played by others, such as tension, friction, gravity, etc. The centripetal acceleration always points to the centre, as does the centripetal force.
So if we consider our object at two points in time, which is undergoing circular motion, our centripetal acceleration is always pointing towards the center whereas the velocity is always tangential or in other words 90 degrees to the acceleration as we see it is here. Our centripetal force can be defined as mass times our centripetal acceleration since force is mass times acceleration. This is also equal to the mass times of velocity squared divided by the radius.
Banked corner. here we see a car on a banked corner tilted at an angle theta the centripetal force is provided by the unbalanced horizontal force in this case it is the horizontal component of the reaction force fr sine theta visualizing this on a vector diagram we have our car in the middle our gravitational force downwards our reaction force upwards at our angle theta since the reaction force is perpendicular to the surface since the car is not moving up or down we know the vertical forces must equal therefore our downwards fg must equal our fr cosine theta which is the vertical component of our reaction force as mentioned above our centripetal force fc is the horizontal component of the reaction force fr sine theta if you're still wondering how we got those components consider that we have a right angled triangle with our right angled there we have our hypotenuse which is our full length of fr our adjacent side is this one here and our opposite is this one here. Using SOHCAHTOA we know that sine of the angle is our opposite over hypotenuse, where our opposite is our horizontal component which is equal to fc and our hypotenuse is fr.
Multiplying both sides by fr we get fc equals fr sine theta which is what we have down here. We can do the same for our vertical component noting that the cosine of the angle is our adjacent over hypotenuse, where our adjacent is our vertical component, which we know is equal to Fg, and our hypotenuse is our FR again. Multiplying both sides by FR, we get that Fg equals FR cosine theta, which is our conclusion that we made here. Newton's law of gravitation.
All objects with mass attract each other. If two objects with masses m1 and m2 are a distance r apart, The gravitational force Fg on each is given by this equation here. Our Fg is equal to our universal gravitational constant, 6.67 times 10 to the minus 11 Newton meter squared per kg squared, multiplied by the product of the masses, divided by the separation squared. Orbital motion. Orbital motion is circular motion where the centripetal force is provided by the gravitational force.
Consider the moon orbiting the earth, at a radius r, our centripetal force is going to be provided by the gravitational force. The equation for centripetal force is this one here, where because we're looking at the motion of the moon, we're using the mass of the moon. The equation for gravitational force is this one here, where we can simplify by dividing our lowercase m out, and by multiplying r to both sides, giving us this equation here, which we can square root to solve for velocity.
Vertical circular motion. A bucket is swung in a vertical circle at the minimum velocity required, as we see here. The centripetal force is provided by the tension force and or the gravity force at different times. At all points our gravity force is pointing downwards, however our centripetal force must always point towards the centre.
At the bottom of our motion our tension force must not only provide the centripetal force. but also counteract gravity. Therefore it's at its largest.
When the bucket is on its way up and on its way down, the gravity force is perpendicular to the centripetal force and therefore has no effect upon it. In this case, the centripetal force is provided by the tension force. At the top of our motion because our bucket is at the minimum velocity required, under these conditions there is no tension force at the top. This means that our downward centripetal force is provided by and equal to our downwards gravitational force. If we consider the energies of the bucket, at the bottom, while it's rising, at the top, and while it's falling, we would see the gravitational potential energy at its minimum at the bottom rising to its maximum at the top, before falling once again, whereas the kinetic energy would be at a maximum at the bottom, since the bucket is travelling its fastest, and at a minimum at the top, where it's travelling the slowest.
And so our kinetic energy is converted to gravitational potential energy, and our gravitational potential energy is then converted back into kinetic energy. Measuring angle. Angle can be measured in either degrees or radians.
One radian is equal to 57.3 degrees. We may also express an angle as a multiple of pi, where one revolution can be thought of as 360 degrees, or alternatively 2 pi. So if a full rotation is 2 pi, then half a rotation must be pi, a quarter pi over 2, and an eighth pi over 4. Translational versus rotational.
Whether an object undergoes translational or rotational motion depends on how a force is applied to it. Pure translational motion occurs when the force points through the center of mass, as it does in these two diagrams. Pure rotational motion occurs when the net force is zero, but the net torque is not. So we see in this situation here, our forces are equal and opposite, so the net force is zero, however because our forces do not act through the center of mass, we get rotational motion because a net torque is produced.
Rotational motion. Angler displacement, which we give the symbol theta, is the change in angle and is measured in radians. Angular velocity, given the symbol lowercase omega, is the rate of angular displacement change, it's measured in radians per second, and can be described with the equation that the angular velocity is equal to the angular displacement divided by the duration. Angular acceleration, given the symbol lowercase alpha, is the rate of angular velocity change, measured in radians per second per second, and can be described with the equation that angular acceleration is equal to the change in angular velocity divided by the duration. Equations of Rotation As with translational motion, we can use kinematic equations to describe rotational motion with constant angular acceleration.
Our kinematic equations are the same, just with rotational values in place of the linear ones. Torque Recall that a force is an influence that acts to accelerate an object. Similarly, a torque is an influence that acts to angrily accelerate an object. It has units of Newton meters.
If we were to apply a force on a wrench at a particular distance, the torque produced is equal to the force times the distance. Recall that a force on a mass causes an acceleration described by F equals ma. We can similarly write a rotational equivalent. with torque our rotational force, rotational inertia which is our rotational mass, and our rotational acceleration, or in other words our angular acceleration.
Rotational inertia. Inertia is the resistance to acceleration. For linear mechanics we call this mass, for rotational mechanics we simply call it rotational inertia.
Rotational inertia has units of kgm2. We're now going to look at some general equations for the rotational inertia of common shapes. A thin ring with a radius r, rotating about this axis, has a rotational inertia of mr squared. If we were to consider a ring with some thickness, with an inner radius of r1 and an outer radius of r2, then our equation becomes half m r1 squared plus r2 squared.
For a solid sphere, Our rotational inertia is 2 fifths MR squared. For a hollow sphere, it's 2 thirds MR squared. For a solid cylinder, we get half MR squared.
And for a thin rod, which is rotating about its long axis, we get 1 twelfth MR squared. It's important to note that all of these rotational inertias are proportional to MR squared. I have here a bike wheel and a length of string. Suspending the wheel from the string and then letting it go, we see the wheel flop down much as you'd expect. Repeating this again but giving the wheel a spin, the wheel remains more or less upright, though due to friction it eventually slows and topples.
So why does the wheel stay up while it's spinning? When the wheel spins it has angular momentum. The larger the angular momentum, the more resistant our angular motion is to change.
And because angular momentum is a vector, Change not only means the magnitude of the angular momentum, but also the direction. Rotational kinetic energy. The kinetic energy of a rotating object is described by EK rot equals half our rotational inertia multiplied by our angular velocity squared.
This should seem suspiciously familiar given that our linear kinetic energy is EK equals half mv squared. where each of these is an analogy of the other. The total kinetic energy of an object is the sum of the linear and rotational kinetic energies, so our total kinetic energy is equal to our linear kinetic energy plus our rotational kinetic energy.
Angler momentum. Linear momentum describes the amount of linear motion an object has, taking into account its mass, or in other words its linear inertia. We describe it with the equation that momentum is equal to mass times velocity.
Angular momentum does this in a rotational context. So our angular momentum, which has the symbol L, is equal to our rotational inertia multiplied by our angular velocity. We can relate angular momentum and linear momentum with each other by considering a mass moving in a circle with a radius R, a velocity v, and an angular momentum about this point of L. Our angular momentum is equal to our mass times our velocity times our radius.
And since momentum is mass times velocity, we can also write this as L equals PR. Here we see a world record figure skater. As she draws her arms in, she reduces her radius, which reduces her rotational inertia.
Because angular momentum is conserved, and our rotational inertia decreases, her angular velocity must increase. Once they're in their spin, the rotational speed is dictated by how tightly they can pull their mass of their arms and their free leg to their axis of rotation. Conservation of angular momentum.
As with linear momentum, angular momentum is always conserved, provided there are no external torques. Consider a main sequence star collapsing to form a neutron star. Our main sequence star undergoes a rotation every 30 days, which is a relatively small angular velocity, but due to its size it has a high rotational inertia. The product of these two is its angular momentum.
When our main sequence star collapses to form a neutron star, the decreased radius causes a reduction in rotational inertia, and because angular momentum is conserved, our angular velocity must increase. and so it's not unusual to find neutron stars rotating in the order of 700 rotations per second. Simple harmonic motion. By definition, a system in simple harmonic motion accelerates towards equilibrium and accelerates proportional to its displacement.
Memorize these two facts very well as it's a common question in the exam. And so if we consider a simple pendulum at two different locations. and our equilibrium is right in the middle.
Our acceleration is always towards equilibrium and is larger the larger the displacement. Representing this on a sine graph, we define our amplitude as the maximum displacement. The simple pendulum.
If we consider a pendulum of a particular length, rotating from equilibrium at a particular angle, more on that later, the period of our pendulum, is 2 pi times the square root of the length divided by the gravitational acceleration. This equation assumes the small angle approximation. This requires that the angle must be small. The equation specifically assumes that sine of the angle is equal to the angle.
And while that might seem like a pretty bold assumption, for relatively small angles, it's not too far off. Therefore, this is only reasonable when the angle is small. At roughly 10 degrees, the error exceeds 1%. and starts to affect values expressed to three significant figures, which is standard at NCR level 3. For this reason at NCR level 3 we put the definition of a small angle as being less than 10 degrees.
I have here a simple pendulum. As you may know the period of a pendulum is only dependent on the length which we can change and the acceleration due to gravity which we typically can't. What it doesn't depend on is the mass.
Keeping that in mind, let's see what happens when we add mass. As you can see, each mass we add decreases the period every time. This happens because the way in which we're adding mass is moving up the center of mass. Since we take the length of a pendulum from the pivot to the center of mass, when we add mass in this way we're effectively reducing the length and therefore reducing the period.
Mass on a spring. A mass on a spring displaced from equilibrium will oscillate with a period described by T equals 2 pi m over k. If we have springs in series, their total stiffness k is going to be the sum of the individual stiffnesses, whereas if we have springs in parallel, then our 1 over k is the sum of our 1 over k constituents.
SHM equations. A system in simple harmonic motion follows a sinusoidal relationship. Note that theta equals omega t, which is just a rearranged version of omega equals theta over t, which is on your formula sheet. We can consider two separate situations. The first is where our system starts at y equals zero, the second is where our system starts at the maximum displacement.
For starting at y equals zero, we use these equations here. Whereas if our system is starting at the maximum displacement, because everything is shifted by 90 degrees, our displacement starts off with a cosine instead of a sine, and we get a different velocity and acceleration equation. The derivation of our velocity and acceleration equations can be done by taking the derivative with respect to time, which if you're familiar with calculus shouldn't be too much trouble, but is not a requirement in this course. Note that the maximum velocity is a omega, and the maximum acceleration is a omega squared. This is the case because the greatest value a trigonometric function can be is 1, and so when our cosine omega t is 1, our velocity is a omega.
A similar situation occurs with our acceleration, the greatest our sine omega t can be is 1, therefore our maximum acceleration has a magnitude of a omega squared. SHM and circular motion. Until now we've represented SHM as a displacement time graph. However, we can also represent SHM as an analog of circular motion called phases. So if we consider a vector with a magnitude A, tracing a circle counterclockwise, we can define its angle to the x axis as theta, and its vertical component as our displacement y.
Here is the situation animated. Energy of SHM. The energy of a system in simple harmonic motion is exchanged between kinetic and potential. So if we consider a graph of our kinetic energy and our potential energy, we see that these are exchanged into each other such that our total energy remains the same. And so this above example assumes that the total energy remains constant, meaning that our displacement would look something like this.
In many systems however, energy is gradually lost. We call this damping.