 This video is probably the last video in the Mechanics series, and what I want to do is to provide an overview, a cheat sheet of sorts that includes all the important equations that you come up in your Mechanics course. If you're studying for your final exam, I would invite you to start writing things before I fill it and think about it yourself. But of course, if you watch this video literally five minutes before your final exam, you might want to just go fast forward and copy down the five screens that I'm going to complete. So first, let's think about the main entities. We had position, we had velocity, and we had acceleration, and we had force, and we had inertia. So what are the letters used in the linear case and what are the units? For position, we usually use S, sometimes you also see XYZ, and the SI unit is meters. For velocity, we use V, a vector, and we use meters per second. And for acceleration, a vector, we use meters per second square. Going on with force, we use F, also a vector named after Mr. Newton, so SI unit Newton. And we had the linear inertia, which is the mass in kilograms. Now what were the angular equivalences of all of those? For position, it's the angle measured in red. Then we had for velocity, we also used the omega in watts per second. For the acceleration, we used the alpha in watts per second square. For the force, the equivalent is the torque in Newton meters. And for the inertia, we have the rotational inertia I, which is measured in kilograms square meters. What are the links between those two? If I want to know the distance traveled along a circle, then I can simply take my angular entity and multiply it by R. If I want to know my linear velocity, I can take my omega and the cross product with the R vector. If I want my acceleration, I have to split it up in my tangential acceleration, which is alpha times R and my centripetal acceleration, which is V squared over R. Then the equivalent for force is torque, which is R cross product F. And then for the inertia, the rotational inertia is the sum of all mass particles times their distance squared from the axis of rotation. So this completes our first page. On our next page, we're going to be looking at momentum. We're going to look at impulse. Then we'll be looking at work, power, kinetic energy, and potential energy. Momentum in the linear case P was calculated as mass times velocity with the unit kilograms meters per second. Impulse was the change in momentum, which was force times time. And its unit is the same kilograms meters per second or newtons times seconds. That's the same thing. Work was calculated as the dot product between F and S as being the distance traveled. So here we get Newton times meters. You see the difference between impulse to change momentum and work to change energy. This time is force times time, and here it's force times distance chopped. Then next one, power P. Don't confuse the momentum P with power P. It's two different entities. It's work over time. And the SI unit is watts. And kinetic energy, we had one half V squared. And for potential energy, we have that in general the change in potential energy is minus the work done by a conservative force. And then we have potential energy of gravity, which is MGH. And we have potential energy of a spring, which is one half spring constant times extension of the spring squared. Now what are the angular equivalences? For the angular momentum, we had L, which is equal to the rotation inertia times omega, which gives us a unit of kilogram square meters per second. Then instead of impulse, the rotational impulse or the change in angular momentum, a vector, is according to this, if you do the equivalence torque times delta T. And for work, if you have something that rotates, so the magnitude of the torque can be calculated as the amount of torque times the delta. Here the torque that is around the same axis as your rotation. So that torque times delta will give you the work. Power is the same definition as before. Nothing changes. For kinetic energy, we can do one half I omega squared. And then potential energy is defined as the same. Gravity doesn't apply. And the spring, if we have a torsion spring, we can do one half torsion spring constant times delta squared. On our third sheet, we're going to look at how we describe how certain things are moving. And we're going to be using kinematics for this. So the third sheet is going to be kinematics. And what we have here is first the basic definitions that velocity is the derivative of position dt and the acceleration is the derivative of velocity. Now for the angular case, this would mean that we have omega is the delta, the deviation of the angle over time and angular acceleration is the omega over dt. Then in both cases, we have for constant or alpha is constant. We have five equations. So we had v as a function of time is v initial plus acceleration times time. Position as a function of time is position initial plus v initial times time plus one half a t squared. And I had position as a function of time as initial plus v final times time minus one half a t squared. Then we had position as a function of time is v initial plus v final over two times time. And then last but not least, v final squared is v initial squared plus two a times s final minus s initial. And then we have the exact same thing for the angular case if alpha is constant. So we have omega as a function of time is omega initial plus acceleration times time. Then we have position as a function of time is omega initial plus theta initial plus omega initial times time plus one half alpha t squared. Then we had the angle as a function of time could also be theta initial plus omega final times time minus one half alpha t squared. Then we have position as a function of time is the average angular velocity. Then we had omega final squared is omega initial squared plus two alpha theta final minus theta initial. On this fourth out of five slides, we're going to be looking at the contributions of Mr. Newton and Mr. Hook. So for Newton, what do we have? We have Newton's first and second law. The first law being a special case of the second one. We're just going to skip the first one and we're going to go to the second one that f net is an A. Then we had Newton's third law of motion. If object one is exerting a force in object two, then object two is exerting a force in object one. Same magnitude but opposite direction. Then we had Newton's universal law of gravity, which is gravitational constant times mass one times mass two over distance squared, which can be simplified on earth to n g. What was the contribution of Mr. Hook? We were talking about a spring. So the force of a spring for Mr. Hook is k x spring constant times the amount of compression or stretching. Now what are the angular equivalences? We have that torque net is equal to rotation inertia times angular acceleration. Again, if angular acceleration is zero, that means the torque net must be zero as well. So we include Newton's first law of motion as well. We can do the torque one. Gravity doesn't have an equivalence. The spring of Mr. Hook actually does. We have a torsion spring. The torque created is proportional to the spring constant times the angle. Now, of our fifth page out of the five pages of the summary, we're going to be looking at conservation laws. So conservation laws. First, we had a vector law which was that the final momentum is equal to the initial momentum plus the change. And the change is by force times time. The angular equivalent to that one is conservation of angular momentum. So conservation of angular momentum L final is initial plus torque times time. Other big conservation law that we had is the same for angular and linear case. And it's actually not a vector law. We have that energy final is energy initial plus the work done by external forces. And if you are considering thermodynamics and chemistry as well, then we could also add the heat as a form to increase or decrease the energy. That concludes my overview of the most important laws in your mathematics course. So wish you good luck for your finals exam and hopefully see you in another video series on another topic.