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Ch 13: Rotational Inertia & EnergyWorksheetSee all chapters
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Ch 01: Intro to Physics; Units
Ch 02: 1D Motion / Kinematics
Ch 03: Vectors
Ch 04: 2D Kinematics
Ch 05: Projectile Motion
Ch 06: Intro to Forces (Dynamics)
Ch 07: Friction, Inclines, Systems
Ch 08: Centripetal Forces & Gravitation
Ch 09: Work & Energy
Ch 10: Conservation of Energy
Ch 11: Momentum & Impulse
Ch 12: Rotational Kinematics
Ch 13: Rotational Inertia & Energy
Ch 14: Torque & Rotational Dynamics
Ch 15: Rotational Equilibrium
Ch 16: Angular Momentum
Ch 17: Periodic Motion
Ch 19: Waves & Sound
Ch 20: Fluid Mechanics
Ch 21: Heat and Temperature
Ch 22: Kinetic Theory of Ideal Gases
Ch 23: The First Law of Thermodynamics
Ch 24: The Second Law of Thermodynamics
Ch 25: Electric Force & Field; Gauss' Law
Ch 26: Electric Potential
Ch 27: Capacitors & Dielectrics
Ch 28: Resistors & DC Circuits
Ch 29: Magnetic Fields and Forces
Ch 30: Sources of Magnetic Field
Ch 31: Induction and Inductance
Ch 32: Alternating Current
Ch 33: Electromagnetic Waves
Ch 34: Geometric Optics
Ch 35: Wave Optics
Ch 37: Special Relativity
Ch 38: Particle-Wave Duality
Ch 39: Atomic Structure
Ch 40: Nuclear Physics
Ch 41: Quantum Mechanics
Intro to Moment of Inertia
More Conservation of Energy Problems
Types of Motion & Energy
Parallel Axis Theorem
Intro to Rotational Kinetic Energy
Moment of Inertia of Systems
Conservation of Energy in Rolling Motion
Conservation of Energy with Rotation
Moment of Inertia via Integration
Energy of Rolling Motion
Moment of Inertia & Mass Distribution
Torque with Kinematic Equations
Rotational Dynamics with Two Motions
Rotational Dynamics of Rolling Motion

Concept #1: Types of Motion & Energy


Hey guys! In this video we're going to talk about how different kinds of motion give you different kinds of energy. I'm going to walk you through a comprehensive list of all the possibilities you might see so that for any kind of problem, you always know what kind of energy goes with the problem, what kind of energy exists in that situation. Let's check it out. What we want to do here is make sure that we know which energies go with a particular situation. A potential problem arise when you have point masses and that's because point masses, if you remember, if they're going in a circular path they have rotational speed. If you have a tiny little mass here, m, and it's going around a path here, it has a distance little r to the middle right and let's say it's spinning that way with an angular speed w. But it also has a tangential speed which is linear. It also has an instantaneous speed that's pointing this way. We call this Vtan. But does that mean, however, that it has linear kinetic energy and rotational kinetic energy? It has a w, so does it have a rotational kinetic energy? It has a V, does it have a linear kinetic energy? Does it have both? The answer is no. It doesn't have two energies. We only have here one type of motion, so we can only have one type of energy. The object only has one type of motion. It only spins around a central point. Vtan is just the linear equivalent of W. Think of it as like a mirror image. If you look at a mirror, there aren't two of you. It's just a mirror is a reflection of you. Vtan is just the linear reflection of w but there's only one velocity, only one motion, one type of motion I should say. If that doesn't make sense yet, that's cool. We're going to do six examples and that's going to cover every possibility. Let's start here. You have a box in a straight line. Does it have linear kinetic energy? Does it have rotational kinetic energy? Does it have both? A box in a straight line, so something like this. The box is moving it has a V, so it has a linear kinetic energy. The box doesn't roll around itself or around anything else so it has no rotational energy, only linear. A disk spinning around itself. A disk, here's the axis in the middle. The disk spins around itself. Does it have kinetic linear? Does it have kinetic rotational? Every time you spin around yourself, you have kinetic rotational. Linear has to do with you moving sideways or up and down, your axis of rotation. I should say your center of mass, the middle of the object has to actually move. If you spin around yourself, the middle never moves so there is no linear kinetic energy in this case. What about the earth spinning around itself? This is a disk. The earth is a sphere roughly. If it spins around itself, it's very similar. There's no kinetic linear and there is kinetic rotation. There's kinetic rotation if you have a shape around itself. But if you're talking about the earth around itself, just that part of the Earth's motion as the Earth spins around itself, it doesn't move sideways. You do know the Earth does move around the Sun but that's a different motion. Here we're talking about just this piece of it. What about the Earth around the Sun? The earth around the Sun, the center of mass of the earth does move. The center of mass of the Earth does move around the Sun. Let's draw that real quick. The Earth is doing this. This is where it gets complicated because you could think of this as the Earth has w around the Sun. It has a w around the Sun but it also has an instantaneous velocity V. You could think of this as linear or rotational. In fact if you solve for it using KL and if you solve for it using KR, you're going to get the same number, the same answer. You could look at the energy either way. The problem is you have to make sure you don't count it as both. What I mean by that is if I ask you for the total energy of the earth going around the Sun, so the kinetic energy of the Earth around the Sun. You can't do KL plus KR. You can't double count it. Here's how I'm going to simplify this. IÕm going to say whenever you have an object spinning around itself or around something else, we're going to call that rotational kinetic energy. We're going to say that there is no linear energy. I just mentioned how you could look at it both ways because there's a V and w. You just can't count it as both. We're going to forget about that. We're just going to make our life simpler and always think of it as rotational energy and never linear energy even though you do have a linear velocity going around this thing. I hope that make sense, around itself, around the Sun. What about the total energy of the earth? I'm going to add a little thing. IÕm gonna call that, I got c and d. IÕm going to call this CD. What about the total kinetic energy of the earth? Kinetic total of the earth, meaning the kinetic energy of the earth around itself plus the kinetic energy of the earth around the Sun. Both of these are rotational. The earth has a rotational energy around itself and it has a rotational energy around the Sun, KR sun. Rotational is if you spin around yourself or if you spin around something else. What about the moon spinning around the earth? What kind of energy does the moon has spinning around the earth? Here you have to know that the earth, the moon, doesn't spin around itself. Here by the way, I want the total energy. What I mean by that is I want to know, does the earth spin around itself and does the moon spin around itself and does it spin around the earth? Moon goes around the earth. LetÕs just do it like this. But you should know that the moon doesn't spin around itself. The moon only has Kmoon. As it's going around the earth, Kmoonself plus Kmoonearth. The moon spins around the earth but it doesn't spin around itself and that's because the moon is locked with the earth. The moon is always spinning facing the same side to us. That's why there's an expression called the dark side of the moon and it's not really dark. It just means that we can never see it because the moon is always looking at us. It's like if you look at a mirror, you can't see your back. You can only see your front. You should know that the moon doesn't spin around itself so it only has a rotational energy around the earth. What about a roll of toilet paper rolling on the floor? You got a toilet paper, it is rolling on the floor. It's running loose right. It has a V and it has a w, but here it actually has two types of motion. Not only it spins around itself but it's also moving sideways so it's doing this. This is called rolling motion and whenever you have rolling motion, you have two types of energy. The total kinetic energy is going to be linear plus rotational so the object actually has both types of energy. This is the only case where you have, out of the six that I've mentioned here, where you have linear and rotational. All the other cases you have either linear or rotational but not both. In the case of the earth going around the Sun and spinning around itself, it has two rotational kinetic energies, one on itself, one around the Sun. That's it. I hope this makes sense. This basically covers every possibility so you should be rocking from here on. If you have any questions, let me know.

Concept #2: Kinetic Energy of a Point Mass


Hey guys! In this video, I'm going to show you how there are two ways to calculate the kinetic energy of a point mass going around a circle. Let's check it out.

Remember if you have a point mass around a circle or in a circular path like this around a distance of little R from the axis of rotation, you have rotational speed _ and you also have a linear equivalent which is our tangential velocity. But you only have one type of motion. All you're doing is this. Your only motion is really rotational motion, so you only have one type of kinetic energy. But you can calculate it using KL or KR. You can use the equation for linear or for KR and that's because these two equations as IÔm going to show you now are equivalents. The most important thing to do here is to make sure you don't double count it. When I ask you for the total kinetic energy of a point mass like this, you can't look at it and say, ÒItÕs got a V so it has a linear kinetic energy and it has a w so it has a rotational kinetic energy. ItÕs got two kinds of energies. Let's add the two of them together.Ó You can't do that because these guys are equivalents. The tangential velocity is basically a mirror of w. It doesnÕt mean there's two. It just means that one basically reflects the other. What you can't do is double count. Let me show you how this works. A small 2-kilogram object, so mass equals 2 kilograms is going around a vertical axis. What is a vertical axis? Remember, axis you can think of it as an imaginary line that you spin around so vertical axis will look like this. It means the object is going around like this. I could draw it like this and the object is doing this. It does this at a rate of 3 radians per second maintaining a constant distance of 4 meters to the axis. This distance to the axis is what we call little r. Little r is 4 meters. I want to know the object's kinetic energy and I want to do this using the KL equation, the KR equation. The purpose of this question is to show you how the answer ends up being the same and I'm going to summarize it at the end. We can do KL which is going to be _ MV^2. Remember that these, V and R, are related by V = R_. What I'm going to do is also write KR = _ I_^2 and I'm going to rewrite one of these equations and you're going to notice how it's going to look exactly like the other. Let's rewrite this one here. Remember, I for a point mass is MR^2 so IÕm gonna replace this with MR^2. I can rewrite _ as well. V = R_, so _ = V/R. Instead of _ here, I'm going to put V/R. Look what happens. This R^2 cancels with this R^2 and we're left with _ MV^2, which is exactly this equation. You can go from one to the other, for a point mass you can do this which means I could have calculated them either way. If I go here, KL equals 1/2 MV^2. Let's get these numbers. W equals 3. V = Rw, so V equals R = 4, w = 3. V is 12. So this is _, mass is 2. They cancel. 12^2. This is 144 Joules. If I wanted to do it using KR, I already showed you how the equations turn out to be the same. I'm just going to plug in numbers differently. If I wanted to do it this way, I could have done _ MR^2_^2. Half the mass is 2, and the distance is 4^2, and the w is 3^2. These two cancel. I have 16*9, which is 144 Joules. If you calculate it using linear, itÕs 144. If you calculate it using rotational, itÕs 144. If I ask you what is Ktotal, the answer is 144. I want you to please write here not 288. You do not add the two. You can get the same answer using the two different equations. To make this simpler for you, I have a convention. I always think of an object going around a circle like this, it has one motion. I always think of this as rotational motion, so I would always do it like this. KL + KR and I would say there's no KL, there's only KR and this will guarantee that you don't double count it. This is just a potentially tricky thing, but once you understand it and get it out of the way, it's never going to bother you again. Let me know if you have any questions.

Practice: The Earth has mass 5.97 × 1024 kg, radius 6.37 × 106 m. The Earth-Sun distance is 1.5 × 1011 m. Calculate the Earth’s kinetic energy as it spins around itself. BONUS: Find the Earth’s kinetic energy as it goes around the Sun.