Ch 10: Rotational KinematicsWorksheetSee all chapters
All Chapters
Ch 01: Units & Vectors
Ch 02: 1D Motion (Kinematics)
Ch 03: 2D Motion (Projectile Motion)
Ch 04: Intro to Forces (Dynamics)
Ch 05: Friction, Inclines, Systems
Ch 06: Centripetal Forces & Gravitation
Ch 07: Work & Energy
Ch 08: Conservation of Energy
Ch 09: Momentum & Impulse
Ch 10: Rotational Kinematics
Ch 11: Rotational Inertia & Energy
Ch 12: Torque & Rotational Dynamics
Ch 13: Rotational Equilibrium
Ch 14: Angular Momentum
Ch 15: Periodic Motion (NEW)
Ch 15: Periodic Motion (Oscillations)
Ch 16: Waves & Sound
Ch 17: Fluid Mechanics
Ch 18: Heat and Temperature
Ch 19: Kinetic Theory of Ideal Gasses
Ch 20: The First Law of Thermodynamics
Ch 21: The Second Law of Thermodynamics
Ch 22: Electric Force & Field; Gauss' Law
Ch 23: Electric Potential
Ch 24: Capacitors & Dielectrics
Ch 25: Resistors & DC Circuits
Ch 26: Magnetic Fields and Forces
Ch 27: Sources of Magnetic Field
Ch 28: Induction and Inductance
Ch 29: Alternating Current
Ch 30: Electromagnetic Waves
Ch 31: Geometric Optics
Ch 32: Wave Optics
Ch 34: Special Relativity
Ch 35: Particle-Wave Duality
Ch 36: Atomic Structure
Ch 37: Nuclear Physics
Ch 38: Quantum Mechanics

Concept #1: Equations of Rotational Motion


Hey guys! When weÕre doing linear motion, you may remember that you had a set of four equations that you would use to solve a whole bunch of different types of problems. In rotational motion, it's exactly the same thing except they're going to take different letters. Let's check it out. As it says here, just like in linear motion, there are four equivalent motion equations for rotation. It's the same exact thing. They just have funny looking letters. As it says here, you often use these when youÕre given a lot of rotational quantities. It's usually a word problem and it starts throwing out things like the velocity, the acceleration and you would use these equations. The process is the same exact one. The equations just look a little bit different. We're going to rewrite these equations real quick. Instead of V IÕm going to have w or Omega, so it's the same thing. w final = w initial. Instead of a, IÕm going to write _t. Same thing here, w final^2 = w initial^2 + 2 ___. Then __ = w initial t + _ _t^2. Then this one is __ = _ (w initial + w final)*t. You can think of this as translating from linear to rotational, same exact stuff, the letters just look different, different variables. I have a star here, an asterisk, because remember same here in some cases your professor may only give you these three equations and want you to stick to three of them. This is the extra 4th equation. You should know by now whether your professor is cool with you using it or not. Remember also that when you're solving motion problems, you need to know three out of five variables. Remember that one variable will be your target and one variable will be your ignored variable. This is the one that will determine the equation to use. This is very straightforward. Let's do some examples. Here a wheel initially at rest. Initially at rest used to be that the initial velocity is zero. It still means that but now its initial angular velocity because this wheel is going to rotate around its central axis. You can think of it as a big disc, something like this. Imagine that's a disc and it has a central axis, meaning like some sort of stick and they can spin around it like that. It starts from rest so the initial w is 0 and it's going to accelerate with the constant 4. Radians per second is acceleration so _ = 4, until it reaches 80 radians per second squared. You can think of this as meters per second, but in rotation. That is your final velocity. It's not actually m/s. You can just think of it that way. w final = 80 radians per second. All the units here are correct. As I mentioned, you can tell you're supposed to use this because you start getting a lot of rotational quantity. In this case, I already know three of them so I know that I can already solve whatever IÕm about to be asked. It says by the time it reaches 80, how many degrees will it have rotated through? How many degrees it's going to have rotated through? ItÕs asking for __, but it wants the answer in degrees which means I'm going to get it in radians because the equations always spit out __ in radians and then you have to convert to degrees. I'm going to do what I always do which is list my five variables here. __ is what we're looking for. The variable out of the five that didn't get mentioned was _t. I'm going to put a little sad face here and IÕm going to pick the only equation out of the four that is missing a _t which is this one. ThereÕs no _t on this one. Same thing as before, w final = w initial and the squares, the squares, ___. __ is what I'm looking for. I'm going to move everything out of the way so __, the target variable, is by itself. wfinal^2 Ð winitial^2, this stuff comes to the other side dividing it. Now we're ready to plug in some numbers. IÕm going to set it up like this, now we're ready to stick the numbers inside of the parentheses. Final velocity was 80, the initial was zero and the acceleration is 4. If you do all of this, you end up with 800 radians. Remember, these equations always spit out radians. Then we're going to convert so I'm going to do ¹ radians at the bottom and then 180 degrees up top. Let me cancel radians with radians. We're left with degrees. 800*180 is 45800 degrees and that's a crap load of degrees. It spins a whole bunch. For Part B, Part B is asking how long in seconds does it take. In other words what is our _t. _t was originally my ignored variable but now we're looking for _t. We can use, since it's the same situation, I can use __. I actually have I know four out of five variables. I only needed three but I know four. When I know more than what I need, it means that I'm going to have more flexibility with the equations. Instead of having to use one specific equation, I can use any equations that have _t which in this case there's three of them. The simplest equation to use would be the first one so I'm going to use that one. We're looking for t, let me circle it. If I move everything out of the way so that t is by itself, it looks like this. t equals, let's plug it in. The final is 80, initial is zero. The acceleration _ is 4, so the answer is 20 seconds. That's it. Very straightforward just like it was before. You just have to basically make the adjustment for the letters and you see different units. It's going to say things like central axis and rotation. It's the same thing just in the rotational world. That's it for this one. Let's keep going.

Example #1: Rotational velocity of disc


Here we have a heavy disc or a very heavy disc. The word very obviously doesn't do anything because it's not a number. A very heavy disc 20 meters in radius, so a disc I wanna draw it like that, radius of 20 meters takes 1 hour to make a complete revolution. The time to make a complete revolution is called period and it's big T. T is 1 hour which is 60*60 seconds or 3600 seconds. Remember, we always convert to the standard units which in this case seconds. It says accelerating from rest at a constant rate. Presumably the disc is rotating around itself because it doesn't say otherwise. It starts with zero, it accelerates at a constant rate. So IÕm gonna write _ = constant but it doesn't tell us what it is so we don't know. We want to know what rotational velocity will the disc have 1 hour after it starts accelerating. After 1 hour or in other words after 3600 seconds, what rotational velocity will be disc have? I'm going to do my little bracket here with my motion variables. Remember, motion variables are the V initial, V final, acceleration, _t and the displacement which in this case is __. I'm missing w initial, I'm missing w final over here. That's what we want to know, what's my final angular velocity. T isn't really one of the five variables so I put it outside. Remember, we're supposed to know three of these things. We know this and this and we got a target. There's two variables here that I don't know. But to solve this problem, I'm supposed to know three. You have to figure out which one you do know here. The idea for this question is that you're supposed to figure out that if the period is 3600 seconds or an hour and I want to know the velocity after that same amount of time, well if it's been a full hour which is how long it takes to make a full revolution, then my __ isÉ Let's see if you can figure this out. What would your delta theta be if it takes an hour to make a full spin and you want to know your __ after that one hour? This would be 2¹ because it's been an hour. An hour is how long it takes to make a full revolution so __ is 2¹. Notice how this wasn't explicitly given to you. It was given to you in a tricky way. Now we know three things and I can solve. This _ here is my ignored variable. Therefore I could go straight into the fourth equation. The fourth equation would work here. Just in case your professor doesn't let you do it with the fourth equation, I'm going to show you how to do it without using the fourth equation. But again, if you could just plug it in and it's going to be really easy. What we're going to have to do is instead of using the fourth equation or use two equations. Why? Because you're going to have to find _ first, and then you're going to have to find w final. If I'm looking for _ first, that means that my ignored variable, while I'm looking for _ is w final. It flips. I was looking for this variable. This one is the ignored. Actually I got to find this first, so this is the ignored. Which equation doesn't have w final? The third equation doesn't have w final. I'm going to go with the equation number 3 and it's going to be __ = w initial t + _ _t^2. We're looking for _. The initial velocity is zero so this is gone and I'm going to move everything out of the way. 2 comes up, __ and the t comes back down over here, _. 2, __ is 2¹ and the time is 3600^2. If you do this, I have it here, you get a very small number. 9.7 x 10^-7 and the reason why the acceleration is so slow is because it took an hour for this thing to complete a full circle. That's the acceleration. Once I know the acceleration, we're now looking for w final. I have four out of five variables which means I'm going to be able to use more flexibility. I'm going to be able to use any equation that has w final in it. I can use the first equation, w final = w initial + _t. w initial is zero. This is just this tiny number, 9.7x10^-7 times time which is 3600 seconds. If you multiply all this, you get 3.5x10^-3 radians per second. That's it for this one. Let me know if you got any questions.

Practice: A tiny object spins with 5 rad/s around a circular path of radius 10 m. The object then accelerates at 3 rad/s2 . Calculate its angular speed 8 s after starting to accelerate.

BONUS: Calculate its linear displacement in the 8 s.

Practice: The turntable of a DJ set is spinning at a constant rate just before it is turned off. If the turntable decelerates at 3 rad/s2 and goes through an additional 30 rotations before stopping, how fast (in RPM) was the turntable initially spinning? 

BONUS: How long (in seconds) does the turntable take to stop?

A wheel with radius 0.20 m starts from rest and turns through 8.0 revolutions in 5.0 s. At t = 5.0 s, what is the radial acceleration of a point on the rim of the wheel?
A 10 cm radius disk is rotating about an axis through its center with a small ant at the rim. From t = 0s to t = 5s, it rotates at a constant angular speed of 5 rad/s. From 5s to 10s, it has an angular acceleration is 15 rad/s2. From 10s to 17s, the disk slows to a stop under constant angular acceleration. (a) At 7s, what is the linear, tangential acceleration of the ant? (b) At 7s, what is the linear, centripetal acceleration of the ant? (c) At 7s, what is the linear acceleration of the ant?
A disk is rotating at 150 rpm. Answer the following questions: (a) What angle, in radians, does the disk rotate through in 25 s if the angular speed is constant? (b) If a brake is applied to the disk, and it takes 5 revolutions to stop, what was the angular acceleration applied by the break, in rad/s2?
A car is initially driving at 20 m/s when the driver sees a dog in the road 100 m in front of him. In order to break in time to not hit the dog, what angular acceleration do the breaks have to apply on the wheels? Assume the wheels have a diameter of 45 cm.
A wheel with radius 0.20 m starts from rest and turns through 8.0 revolutions in 5.0 s. At t = 5.0 s, what is the tangential acceleration of a point on the rim of the wheel?
At t=0 a grinding wheel has an angular velocity of 26.0 rad/s . It has a constant angular acceleration of 31.0 rad/s2 until a circuit breaker trips at time t exttip{t}{t}= 2.10 s. From then on, it turns through an angle 436 rad as it coasts to a stop at constant angular acceleration.a) Through what total angle did the wheel turn between t=0 and the time it stopped?b) At what time did it stop?c) What was its acceleration as it slowed down?
An airplane propeller is rotating at 1910 rev/min.a) Compute the propellers angular velocity in rad/s.b) How long in seconds does it take for the propeller to turn through 36°?
A CD-ROM is accelerated from rest with constant angular acceleration αo.Find the time for the disk to complete the first full loop. [a] t = 1/2  (αo2 ) [b] t = 2π/αo [c] t = 4π/αo [d] t = (2π/αo)1/2 [e] t = (4π/αo)1/2
A fan is turned off, and its angular speed decreases from 10.0 rad/s to 6.3 rad/s in 5.0 s.  What is the magnitude of the angular acceleration of the fan?A) 0.37 rad/s2B) 11.6 rad/s2C) 0.74 rad/s2D) 0.86 rad/s2E) 1.16 rad/s2
A wheel accelerates from rest to 59 rad/s at a rate of 74 rad/s2. Through what angle (in radians) did the wheel turn while accelerating? A) 24 rad B) 30 rad C) 19 rad D) 47 rad
A wheel with radius 0.200 m starts from rest at t = 0 and then starts to rotate with constant angular acceleration about an axis at its center. At t = 5.0 s the wheel has turned through 4.00 rev. What is the angular acceleration of the wheel, in rad/s2?
A wheel with radius 0.200 m starts from rest at t = 0 and then starts to rotate with constant angular acceleration about an axis at its center. At t = 5.0 s the wheel has turned through 4.00 rev. At t = 5.0 s, what is the magnitude of the linear velocity of a point on the rim of the wheel?
A large wheel of radius 0.300 m is initially at rest and then starts to rotate with a constant angular acceleration of 0.400 rad/s2 about an axle at its center. What is the tangential velocity of a point on the rim of the wheel after the wheel has turned through 1.60 rad?
A wheel of radius 0.300 m is mounted with frictionless bearings about an axle through its center. A light rope is wrapped around the wheel around the wheel and a block is suspended from the free end of the rope. When the system is released from rest, the block descends with a constant linear acceleration of 4.00 m/s2.What is the angular  velocity (in rad/s) of the wheel after it has turned through 5.00 rev?