Hey guys. Hey Bo. Hi Bo.
Flipping Physics Ladies and gentle people, the bell has rung, therefore class has begun. Therefore, you should be seated in your seat ready and excited to start class. You should be aware that today we're going to be reviewing everything we've ever learned in mechanics in AP Physics C. So let me hear you get excited. Oh wow.
Oh yeah. Please understand that today, I just want to move as quickly as possible. We're going to review. I'm not going to call on anybody.
I'm not going to ask any questions. That's just the way it needs to be today, okay? I don't know if I can do that.
No problem. I can do that. Here we go. The basics.
Vector versus scalar. It may seem very elementary, and it is. A vector has both magnitude and direction, and a scalar has magnitude only. However, it comes up all the time.
Just one basic example comes up on multiple choice questions. Very often that momentum is a vector, and they do some sort of conservation momentum problem, and people forget that it's a vector, and they just treat it as a scalar. So please remember the basic concept of a vector versus a scalar. Next, the concept of uniformly accelerated motion. Uniformly accelerated motion.
The basic idea that the acceleration, that when the acceleration is uniform, in other words, the acceleration equals a number, that there are a basic set of equations that you can use. There are only three equations given on the equation sheet. However, there are actually four.
I'm going to write them out. Mr. Again, there are four UAM equations. I don't know why this last one they choose not to give you on the equation sheet. You're more than welcome to use it. And you'll notice I have a slightly different form, but this is the form I prefer them to be in.
Again, this is UAM, uniformly accelerated motion. You can use this whenever the acceleration equals a number. In other words, the acceleration does not change.
It is constant. U. Now, there are five UAM variables, there are four UAM equations.
If you know three of the variables, you can figure out the other two, and this leaves you with one happy physics student. Next, we're going to talk about acceleration. You can see I've actually written two equations for acceleration.
We have acceleration equals the derivative of velocity as a function of time and acceleration equals the change in velocity over change in time. The derivative would be the instantaneous acceleration. The change in velocity over change in time would be the acceleration over a time period which would be the average acceleration. Please remember that with any derivative, you can actually rearrange it to get an integral.
Mr. In other words, velocity is also equal to the integral of acceleration as a function of time. We can do the same thing with velocity, where we have the equation for instantaneous velocity, the derivative of a position as a function of time, and the average velocity, the change in position over change in time. And so we can rearrange the derivative to get that the position is equal to the integral of the velocity with respect to time.
Also, you need to remember what the derivative and the integral actually mean. The derivative means literally the slope of the line and the integral means the area under the curve. Now, I put under in quotes because what that means is the area between the curve and whatever's on the x-axis.
And in fact, one of the things you also need to remember is that area Under, or below the x-axis is actually negative. So under the curve specifically means that the area between the line and the x-axis where above the x-axis you have positive area, below the x-axis you have negative area. Mr. You guys do realize he posts all of his lecture notes at FlippingPhysics.com, right?
Yes, I know that. However, I find that I learn it better and I remember it when I write it all down. Thank you. Suit yourself. Next, let's talk about the basic concept of projectile motion.
Mr. Projectile motion is a basic idea when you have an object flying through the vacuum that you can breathe in two dimensions, which means that in the x direction, we don't have any forces, so the net force is equal to zero in the x direction, therefore the acceleration is equal to zero, therefore you have a constant velocity, and you can use the equation for constant velocity in the x direction. In the y direction, you know that the acceleration in the y direction equals negative g, where g on planet Earth is a positive 9.81 meters per second squared. This is uniformly accelerated motion in the y direction.
In fact... Mr. It's actually called free fall in the y direction. And notice that the change in time, which is the only variable which is a scalar in these equations, is therefore independent of direction.
And usually what you're doing is you're solving for the change in time in one direction and then applying it to the other direction. The net force or the... Some of the forces equals mass times acceleration where both force and acceleration are vectors. Remember, whenever you're summing the forces, you have to draw a free body diagram. A free body diagram.
Mr. Or FBD for Free Body Diagram. Sometimes you also call it a force diagram. Whenever you sum the forces, you have to identify the object or objects you're summing the forces on and you need to identify the direction.
And if the direction isn't clear, what's positive, you also need to identify what the positive direction is. Mr. Now, an interesting thing about Newton's Second Law is this is not quite Newton's Second Law. A better approximation of Newton's Second Law actually has to do with the derivative. This is Newton's Second Law using the derivative, where the net force is equal to the derivative of momentum as a function of time, where again, force and momentum are both vectors. What this means is where momentum is mass times velocity, A more accurate representation of Newton's Second Law is that the net force is equal to the derivative of mass as a function of time times velocity, plus the mass times the derivative of the velocity as a function of time.
One of the things that we usually assume is that the mass of the object is not changing, and therefore this whole piece will go to zero, and you just end up with the net force equals mass times the acceleration. The next basic concept is the concept of impulse. The symbol for impulse is a capital J. Impulse is the integral of force with respect to time and it's also equal to the change in momentum. And again, let's just review the fact that the impulse then is the area underneath the curve because it's the integral of the force with respect to time.
And it's not unusual to see some sort of multiple choice question that has to do with, it gives you a graph and you have to figure out the impulse or the change in velocity or something of that sort. And you just need to figure out the area under the curve. Sometimes they also do it with work. Returning back to our net force equation equals the derivative of momentum as a function of time.
If the net force on the system is equal to zero, that means the derivative of the momentum as a function of time is equal to zero, and that means that the momentum does not change as a function of time. Mr. In other words, you end up with conservation of momentum. The sum of the initial momentum equals the sum of the final momentum, again, where momentum is a vector.
And so that is where all the forces are internal, that it means that the sum of the forces equals zero, and therefore, you have conservation of momentum. Let's talk about specific forces. We could talk about the force of friction, for example.
The force of friction by definition is less than or equal to mu times the force normal where mu is called the coefficient of friction and does not have any dimensions and is dependent on the two different surfaces that are interacting. We actually have two different types of friction or two different types of coefficients of friction. We have static which is non-moving friction and kinetic which is moving friction. So it ends up, we have several, a couple of different equations here. In other words, because we can have both kinetic and static friction, we have that the force of kinetic friction is equal to mu k times the force normal, when the force of static friction is actually less than or equal to the coefficient of static friction times the force normal.
However, usually you end up using the maximum force of static friction, which is equal to mu s times force normal, but it is important to remember that the force of static friction actually adjusts up and down to prevent the object from moving. Some basic ideas about the direction of the force of friction. What?
One, it opposes motion, meaning it tries to prevent the object from moving, or if it is moving, is opposite the direction of the motion. That's not quite always true in AP physics. You can have an example for if you have a box on a truck and the truck is accelerating, there's a force of static friction on the box that's in the back of the truck, and that force of static friction would actually be pushing the box forward.
That's sort of an unusual case, but for most of the time, the force of kinetic friction would be opposite the direction of motion, and the force of static friction would oppose what direction the object is trying to move. The force of friction is always parallel to the surface, and one of the things that I always add is that the force of friction, the direction of it, is independent of the direction of the force applied. Because so often students simply think the force of friction is opposite the direction of force applied, which is not true. Mr. Work. Work is equal to the integral of force, dot product with respect to the position.
If you have a constant force, that means that the force is just the dot product of the force and the position or the r vector, and you could just do F r cosine theta, the force times the change of position times the cosine of the angle between the force and the change in position. Next, Mr. We have three different kinds of mechanical energy. Mr. Kinetic energy, one half mass times velocity squared.
Gravitation potential energy, mass times the acceleration due to gravity times h, the vertical height above the zero line. Please make sure you always set the zero line before you do a problem. Elastic potential energy, one half kx squared, where k is the spring constant and x is the displacement from equilibrium position.
Three different kinds of mechanical energy. Three different equations that have to do with mechanical energy. Mr. First equation is conservation of mechanical energy where mechanical energy initially equals mechanical energy final. With all three of these equations, please always identify your initial and final points. If you're going to use conservation of mechanical energy, you have to have no force applied and no force of friction.
or at least no energy converted to heat and sound, etc., via the force of friction. For the work due to friction equals change in mechanical energy, that you need to have no force applied in order to use that equation. And the network equals the change in kinetic energy is actually always true. And just so you know, these two equations often get confused.
The work due to friction equals the change in mechanical energy, whereas the network equals the change in kinetic energy. Be careful to use, to remember those correctly. Mr. Power, which is the derivative of work with respect to time, which is equal to the derivative of the quantity for work, which is the dot product of force and position with respect to time, or the derivative of position as a function of time is velocity, if you have a constant force, it would just be the dot product of force and velocity. Power is the rate at which work is being done, the joules per second. Please remember that you can rearrange this equation again.
Where then work would be equal to the integral of power with respect to time. There is an equation that is always true whenever you have a conservative force. A non-conservative force would be friction, the force of friction, and a conservative force is a force for which the path does not matter, where the work done is independent of the path. So if you have a conservative force, this equation, which is not on your equation sheet and often comes up, you need to have memorized, again, for a conservative force.
For a conservative force, that force is equal to the negative derivative of the potential energy associated with that force with respect to position. Mr. An example. For the force of a spring, which is equal to the negative derivative of the potential energy associated with that spring with respect to position, that's going to be equal to the negative derivative of 1 half kx squared with respect to position.
So the derivative of that ends up being equal to negative kx, which you know. is the force of a spring, just as an example. Center of mass.
The equation for the position center of mass for a system of particles would be mass one times position one plus mass x2 times position 2 plus however many you have divided by the sum of the masses, mass 1 plus mass 2, however many we have, where x1 and x2 are the position relative to some reference line which you get to choose. Again, that's for a system of particles. And you can do it the x center of mass, you can do the y center of mass, you can do the r position center of mass, which would be in three dimensions.
You can even take the derivative of this to get the velocity of the center of mass. You can take the second derivative of it to get the acceleration of the center of mass, just as examples. Again, for a system of particles.
If instead you have a rigid object with shape, the equation is the integral. For that we have the r center of mass, or position center of mass, is equal to one over the total mass times the integral of r dm, so the position with respect to mass. Lecture notes are available at FlippingPhysics.com.
Please enjoy lecture notes responsibly.