chapter four dynamics Newton's laws of motion so dynamics is the study of the source of motion and the source of motion is force so in Chapter four we're gonna study force will study Newton's three laws of motion we'll discuss the differences between mass and weight and then we'll talk about problem solving approaches and problem solving strategies the force is any sort of push or pull on an object for an object to change its velocity a force must act on the object we would say force is a vector because force has both magnitude and direction in the macroscopic state looking at objects the way we look at them we can distinguish between contact forces and field forces contact forces require some sort of physical contact and field forces act through empty space now when we when an object is in equilibrium we say there's no net force acting on the object and we could say that the object is either at rest or is moving with constant speed so in other words the object is not accelerating in the microscopic world we talked about four fundamental forces of nature we have the gravitational force which has to do with objects that have mass the electromagnetic force has to do with objects that have charged the strong nuclear force which is associated with the nucleus of atoms and the weak force which is associated with radioactive decays now all of these are field forces in the microscopic world we don't really discuss contact force as a matter of fact most contact force that we would discuss in the macroscopic world tend to be electromagnetic in nature in the microscopic world this would be things like the normal force the reason why I don't fall through the floor as I stand on the floor is because the floor pushes back up on me and the direct force between my and the floor has to do with the layer of electrons and the negative negative repulsion of the electrons on the bottom of my shoes and the electrons that are composing the floor itself now Newton's first law didn't first law is often called the law of inertia inertia is that property of an object that will describe its resistance to change in motion so oftentimes we can discuss the measurement of inertia as being the mass of an object okay we will talk about this in a little bit Newton's first law of motion can be stated in several ways one simple way to state Newton's first law is objects at rest tend to stay at rest objects in motion tend to stay in motion unless acted on by a net force or we could say more specifically every object in an inertial reference frame continues in its state of rest or of uniform velocity in a straight line as long as no net force acts on it in the case of this book here if the person holding the book pushes to the left and there's some frictional force on the table acting to the right and since those forces are vectors if they have the same magnitude and opposite direction they will add to zero then the book will just sit on the table and remain motionless so even in ancient times intuition led people to believe that some sort of action was required to keep objects in motion but we realize now with our better understanding of the universe that there exists resisting in frictional forces that actually act on objects in motion to cause them to slow down if we were in an environment that where there was no friction or there were no resisting forces objects in motion would simply continue to stay in motion unless another force were to act on them let's look at this conceptual example a school bus comes to a sudden stop and all of a sudden the backpacks on the floor start to slide forward what force causes them to do that well it turns out no force causes them to do that the objects are already in motion and the Amish will continue to stay in motion unless stopped by friction or some collision so if an observer on the outside of the bus were to watch the bus along with the backpacks be in motion as the bus begins to stop the backpacks which are items themselves continue in motion unless the friction force from the floor of the bus stops them or slows them or the backpack slide and collide with the walls of the bus or the walls of the seats so that brings us to to try and understand our observers we would say that Newton's first law of motion only really works if our observers are in inertial reference frames an inertial reference frame is any reference frame that's not rotating or accelerating in some way now mass mass is the property of an object that specifies how much resistance that object exhibits to changes in its velocity or in other words mass is a measure of an object's resistance to change in motion in the International System SI system the mass the mass property is measured in kilograms now mass is not weight okay weight is mass times the acceleration due to gravity which on earth we know is G and G equals 9.8 meters per second squared mass is scaler weight which is force is a vector these are very different things your mass is an intrinsic property its inherent it doesn't change as you change your position in the universe weight however depends on the dominant body on which that object with mass M exists and if the object of mass M exists on a body that exhibits a gravitational force an acceleration due to gravity G then the weight of the object of mass M is M times G so on the moon it's understood that the acceleration due to gravity is about one-sixth the acceleration due to gravity on earth so if you go to the moon your mass stays the same because the intrinsic property the inherent property of your resistance to change in motion doesn't change the amount of substance that composes you doesn't change however on the moon the acceleration due to gravity is less so your weight would be less now moving on to Newton's second law Newton's second law can be easily stated F equals MA where F is the net force acting on object of mass M and a is the acceleration notice that this is a vector equation F equals M a so for vector equations we know that in three dimensions we would have three component equations so we can see F x equals MA X FY equals MA y and F Z equals M a Z these are our component equations now don't forget when we have component equations which are composed from vector equations what what couples all of these component component equations together are the scalars and in this case that's the mass now because for F equals M times a we can look at the units for M which is kilogram the units for a acceleration which is meter per second squared and when we multiply those we get kilogram meter per second squared which is Newton so the SI unit for force is Newton so looking back at the at the mathematical statement of Newton's second law F equals MA we could say that our understanding of that equation tells us the following translation the net force acting on an object is proportional to the acceleration of the object and the proportionality constant is the mass for higher mass a net force would equal net forces would produce lower accelerations so let's look at this example we want to calculate the net force needed to accelerate a a 1000 kilogram car and one-half G and B a 200 gram Apple at about the same rate so we'll use F equals MA Newton's second law and we would see first we need to find the acceleration the acceleration is one-half times g g is 9.8 meters per second squared half of that gives us four point nine meters per second squared so the net force required to accelerate a 1000 kilogram car is 1000 kilograms times four point nine meters per second squared or about 5,000 Newtons four thousand nine hundred Newton's for the Apple changing two hundred grams to kilograms we get point two zero zero kilograms times the same acceleration gives us about point nine eight Newton's or around one Newton now let's look at this example what average net force is required to bring a 1500 kilogram car to rest from a speed of 100 kilometers per hour within a distance of 55 meters assume constant operation so remember assuming constant acceleration can allow us to use our kinematics problems it can allow us to do different things so let's solve this problem first we'll write down our Givens so we know that the initial velocity is 100 kilometers per hour which with a little bit of analysis we could find that to be twenty seven point eight meters per second we want this car to come to rest so V final is zero we'll set the cars initial position at x equals zero and we'll see that it travels 55 meters before coming to rest so that will be X final using our kinematics equation V squared is V initial squared plus 2ax minus X initial if I wanted I could put X subscripts on the v's but using our kinematics equation we can easily solve for the acceleration with a little bit of algebra subtract V initial squared / divided by two divided by Delta X and I end up with a is negative seven meters per second squared being negative means that since my initial velocity is positive positive to the right my acceleration is negative to the left and that causes my object to slow down then to find the net force I just simply multiply M times a so the mass of the car the object slowing down times the acceleration gives me negative 11,000 Newton's as the for the average force required to slow this car from 100 kilometers per hour to rest in a distance of 55 meters now back to weight versus mass let's discuss the gravitational force this is the attractive force exerted by a massive object on another massive object so when you're near Earth we know that the acceleration due to gravity is 9.8 meters per second squared and the gravitational force that Earth Zana mass M would be mg and it always points towards the larger object we'll discuss in a little bit that the smaller object also would attract the larger object and these two objects would interact by means of forces now if two objects are at the same location meaning the same distance from Earth then their acceleration due to gravity G is the same and what that tells us is that their ratio of weights equals the ratio of masses because if weight is mg then the weight one is m1 G and the way to is m2 G and if I divide these two equations the ratio of weights equals the ratio of masses so what this tells us eventually is that we can discuss mass in two different ways one the original way we discussed mass was the inertial mass the resistance to change in motion and here a component of the weight of an object we call the gravitational mass well it turns out and we'll show this in future semesters it turns out that gravitational mass and inertial mass are the same thing there's no reason to think that they're different another force to discuss is the normal force when an object is at rest there must be no net force so if an object exists on earth it must have weight so an object at rest on a table must have some force countering that weight and here we can see here we have a statue it's on the table the gravitational force is pulling downward and then the normal force from the table is pushing upward and that net force as to zero that's why the object remains at rest here let's do this example a friend has given you a special gift box a box of mass 10 kilograms with a mess siree surprise inside the box is resting on the smooth horizontal surface of a table determine the weight of the box and the normal force exerted on it by the table that's part A Part B now your friend pushes down on the box with the force of 40 Newtons again determined the normal force exerted on the box by the table and then Part C if your friend pulls upward on the box with the force of 440 Newtons what now is the normal force exerted on the box by the table let's do this problem so Part A since we understand that the box is always at rest the total force must be zero in other words the net force on the box must add to zero and this is vector sum so I must have forces in opposite directions so here I can see that the normal force equals the weight which is ten kilograms times 9.8 m/s^2 so here I have the Box Part A the weight is down the normal force magnitude must equal the weight but the force vectors must be opposite in direction so whatever the normal force magnitude is it has to equal mg that's why we end up understanding the normal force equals the weight in Part B now we have two downward forces we have the weight plus 40 Newtons so the table must push up with a total force equal to the total downward force so now we know that the normal force magnitude must be 98 Newtons which is the weight plus 40 Newtons which gives us 138 Newtons now what do you think will happen if I reduce some of the downward force by lifting upward a little bit is that going to increase or decrease the normal force well it turns out that if I reduce some of the downward force by pulling upward the normal force then reduces so if I look at the box and I say okay my normal force is unknown what's all the force on the Box other than the normal force I have the same weight 98 Newtons down we then add up all the forces than the normal force noting that the weight is downward and the upward force is upward then the sum of those two forces is the weight 98 Newtons minus 40 Newtons or 58 and and then we would see that the normal force has to balance the 58 Newtons downward with 58 Newtons upward let's do this example what happens when a person pulls upward on the Box in the previous example with a force greater than the boxes weight let's say 100 Newtons so here I have this this box now with a net force upward because I can see the upward force is greater in magnitude than the downward force so the total force is the sum of FP which is up and mg which is down and when I add those two vectors I end up subtracting their magnitudes because their opposite direction so 100 minus 98 is two Newtons and I see that the net force is two Newtons upward so from Newton's second law F equals MA if I have two Newtons upward then the acceleration must be two Newton's divided by the mass upward so if this box is ten kilograms then the acceleration is two Newton's divided by ten kilograms which is 0.2 m/s squared