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: Rotational Velocity & Acceleration


Hey guys! Now that we've seen how rotational position and displacement relate to linear position displacement, we're going to look into velocity and acceleration. Let's check it out. The rotational equivalence of linear velocity and linear acceleration are rotational velocity and rotational acceleration or angular velocity and angular acceleration. Similar to how x becomes _ and delta _x becomes __ in rotation, V and a will take different letters as well. Average velocity if you remember is _x / _t and the units are meters per second. Average acceleration or I should say acceleration is _V / _t, change in velocity over change in time, and it's measured in meters per second squared. That's if you are dealing with linear problems. If you have rotational problems, angular motion problems, instead of V we're going to use w. w (omega) is a Greek w, so it's like a curly w. ItÕs essentially w. Instead of _x / _t, remember instead of _x we now __. W is __ / _t. V is how quickly I can get from here to here. It's a measurement of how quickly I move between two points. w is a measurement of how quickly you spin in a circle. Remember also that this was in meters and this is in radians. Instead of meters, you have radians. Instead of meters per second, you're going to have radians per second. The acceleration is very similar. Instead of a we're going to have _ which is a Greek a, and same thing here, acceleration is velocity over time. Acceleration would be angular or rotational velocity over time, so itÕs __ / _t, and it's radians per second squared. You might start to see a pattern. The variables are x or _x, and then V and a. They become _, w and _ and the pattern is that English letters are representing linear motion and Greek letters are represent the angular or rotational motion. These are all Greek letters Ð _, _ and _. _ is a way to indicate how quickly something spins. There are actually three additional variables that will help us describe how something moves and they're related to w, in fact theyÕre all related to each other. You might remember I showed you right here we just did w is __ / _t. You might remember that we talked about if you have a complete revolution, then your __ is 2¹. The entire angle for a complete revolution is 2¹. The time that it takes for a full cycle is called period t. One way that you can rewrite w is not just __ / _t, but also 2¹/t. Remember also that period and frequency are inverse of each other. Instead of 2¹ / t, I could also write this as 2¹f. You have w is a measurement of how quickly something spins, period is a measurement of how quickly something spins and frequency is also a measure of how quickly you spin and they're all related by this equation. Last one we're going to talk about is rpm. rpm stands for revolutions per minute so 1 rpm is 1 revolution per minute. A Hertz which is a unit of frequency, frequency is measured in Hertz is one revolution per second. You can see how these two are related. For example if I tell you something spins with 120 rpm or at a rate of 120 rpm, rpm is simply 120 revolutions per minute and what I can do is I can say I can put a minute up here and convert this into seconds by divided by 60. One minute equals 60 seconds, I can do this. Look what I end up with. 120 divided by 60 is 2. I put 2 revolutions per second. Revolutions per second is frequency. Revolutions per second right here is frequency. If you have rpm, you can convert to frequency by dividing by 60. I'm going to quickly write another equation here which is that frequency is rpm over 60. We have a way to connect all of these guys. w, T, f and rpm are all connected. Typically what you're going to do is we're going to convert from any of these 3 Ð T, f or rpm Ð back into w using these equations. The idea you see more of this later is most of the equations I give you will have w but not any of the other letters. I have a little diagram to connect all of these things. If you have rpm, you will want to convert it to frequency. The way you do this is by using f = rpm / 60. That's how you go from rpm to frequency. Then from frequency, you can convert into either period using the fact that period is the inverse of frequency or you can convert into w using the fact that w is 2¹f. Obviously you can convert it backwards in any direction. Generally you want to end up here, but you might have to go from let's say w to rpm. We'll do some of this stuff. These are the 4 units to tell you how quickly something spins and you may have to convert among them. IÕm trying to highlight this. There you go. One last point before I do an example and I've mentioned this briefly before. Rotational equations which is what I'm showing you will feel already by now. They work for two different situations Ð one is when you have a point mass. A point mass is a tiny object of negligible size that spins around the circular path. We call it a point mass because we just represented it by a point. It has no volume. I'm going to say that the radius of this object is 0. Imagine a sphere with radius. It has no radius. It has no volume. Radius is zero. You can also write if you want volume = 0. IÕm actually gonna write out volume so you don't think it's velocity. It's a point. It could be a small object that we simplified down to a point, so that's point mass. The other one is when you have a rigid body which is something where the radius is not zero. The radius is greater than zero so it has volume. I refer to this as either a rigid body, that's sort of the classic textbook name, or a shape. The reason why I like to think of it as a shape is because in these problems usually we'll be told what the shape is. If I tell you have a small object, that's a point mass, and if I tell you have a cylinder usually I'll tell you that it's what the radius is and then you treat that a little bit separately. You can either have a point mass, IÕm gonna draw a tiny little M going around a circle. The circle has a radius r, in this case little r is the distance from the to the edge of the circle and you are going around at the edge. R is the radius of the circle and then little r is how far you are from the center. Those are the same thing or you can have like a cylinder for example spinning around itself. Let me draw a little cylinder here, it looks like a cake, and a cylinder that rotates around itself and that cylinder has a radius of R. You can have either one of these two situations. We'll look at both a lot. That's a quick intro. I gave you some equations how to connect things together between these four different variables and weÕre now gonna do a problem. I have a 30-kilogram disc, so mass = 330, radius = 2. Let's draw a disc, a circle, the radius is 2. This is a disc. It's a rigid body or a shape. The radius of this thing is 2 and it rotates at a constant 120 rpm. I'm going to write here that rpm equals 120. We want to know its period and angular speed. For Part A, I want to know what is big T and then for Part B I want to know what is w, which is w, angular speed same thing angular velocity same thing as rotational velocity or rotational speed. How do we tie it together? If you look at this little diagram here, we can convert from one to the other. I'm going to convert rpm into frequency and then I'm going to convert from frequency into period and into w. First weÕre gonna go frequency = rpm/60, so it's 120 divided by 60 is 2. The units are Hertz but I don't actually want frequency, I want period. Period will be 1/frequency, _ = 0.5. Period is measured in seconds. To find angular speed w, I just used that w is 2¹ / T or I can use that w 2¹f. I'm going to use 2¹f because it's more straightforward. 2¹ frequency is 2, so I get 4¹ and if you multiply all of this, the answer is 12.6 radians per second. That's it for this one. Let's do the next one.

Example #1: Rotational velocity of Earth


Here we have an example. WeÕre calculating the rotational velocity of the earth as, in two different situations, as it rotates around itself and rotates around the Sun. LetÕs look at the first one, Earth's around itself. Actually I want w of Earth around self. w is either __ / _T or 2_ / t or 2_f. As the Earth rotates around itself, what's the __, what's the time? I'm not telling how many degrees it's going through. I'm not telling you this either. One of the things you can do is you can then use some information you know about the earth. You know how long the period of the earth is, how long here it takes to make a complete revolution around itself. If itÕs a complete revolution around itself, that takes 24 hours. It takes 24 hours, so all I can do is I can just write w = 2_ / 24 hours. Obviously I can't just leave it as 24 hours, I have to convert that into seconds. It's going to be 2_ / 24*60*60 and you multiply all of this, you get that the w for the earth is 7.27 x 10^-5 radians per second. It's a very small number, and that's because the earth takes a very long time to spin around itself relative to like a DVD or something. B, what is the rotational velocity of the earth rotating around the Sun? Very similar problem. w equals, I can use any of these equations. In this case what I'm going to use is again 2_ / T, and the period of the earth going around the Sun is 365 days. You don't have to worry about leap years or any of that stuff. Just simplify it by doing this. It's 2_ / 365. It's basically all of this, this is one day, 24 hours. It's all of this times 365. I'm going to repeat the bottom there. This is going to be an even smaller number. I get 1.99 x 10^-7 radians per second. The last thing I want to point out to you guys is that if the earth is rotating around itself, in this case we're talking about a rigid body that's spinning around itself because it's sort of a sphere not exactly but you could approximate it as a sphere that spins around itself. The Earth rotating around the Sun actually acts as a tiny point object. Why? Because relative to the size of the Sun and relative to the distance between the two, the earth is so tiny that you would consider it a point mass going around the circle. Here you have a rigid body spinning around itself. That's an important distinction in these two examples. An object can depending on the situation be thought of as a rigid body that spins around itself or a tiny point mass that spins around a circular path. That's it for this one. Hopefully it makes sense. Let me know if you guys have any questions.

Practice: Calculate the rotational velocity (in rad/s) of a clock’s minute hand. 

EXTRA: Calculate the rotational velocity (in rad/s) of a clock’s hour hand.

Practice: A wheel of radius 5 m accelerates from 60 RPM to 180 RPM in 2 s. Calculate its angular acceleration.

As a particle with a velocity v in the negative x direction passes through the point (0, 0, 1), it has an angular velocity relative to the origin that is best represented by vector A. 1 B. 2 C. 3 D. 4 E. Zero
A wheel of radius 0.5 m rolls without slipping on a horizontal surface. Starting from rest, the wheel moves with a constant angular acceleration of 6 rad/s2 . The distance traveled by the center of the wheel from t = 0 to t = 3 s is about: A. none of these B. 2.1 m C. 13.5 m D. 18 m E. 27 m
When you look up into the sky, you always see the same part of the moon, no matter what time of the month or year it is. In order to achieve this, the rotational period of the moon must be equal to its orbital period (how long it takes to orbit the Earth). Given this fact, what is the angular velocity of the moon due to its spinning about its own axis?
When you ride your bicycle, in what direction is the angular velocity of the wheels? A) to your left B) to your right C) forward D) backward E) up
Two solid discs are rotating about a perpendicular shaft through their centers, as shown in the figure. Disc A, has a radius that is twice as large as disc  B,? Which of the following statements is NOT true? A) A point on the rim of disc  A, has twice the linear speed as a point on the rim of disc  B. B) The direction of the angular velocity is to the right. C) Every point on the body has the same angular acceleration. D) The linear acceleration of a point on the rim of disc  B is the same as the linear acceleration of a point halfway from the center to the rim on disc  A. E) The angular velocity at a point on the rim of disc  A is twice the angular velocity of a point on the rim of disc B.
What is the acceleration experienced by the tip of the 1.6 cm -long sweep second hand on your wrist watch?
A fan blade rotates with angular velocity given by ωz(t) = γexttip{gamma }{gamma} - βexttip{eta }{beta}t2.a) Calculate the angular acceleration as a function of time.b) If γexttip{gamma}{gamma_1} = 5.05 rad/s and βexttip{eta}{beta_1} = 0.805 rad/s3 , calculate the instantaneous angular acceleration αz at t exttip{t}{t_0}= 3.10 s .c) If γ exttip{gamma}{gamma_1} = 5.05 rad/s and βexttip{eta}{beta_1} = 0.805 rad/s3 , calculate the average angular acceleration αav - z for the time interval t = 0 to t exttip{t}{t_0}= 3.10 s .
A roller in a printing press turns through an angle θ(t) given by θ(t) = γt2 - βt3 , where γ = 3.20 rad/s2 and β eta= 0.500 rad/s3{ m{ rad/s}}^3.a) Calculate the angular velocity of the roller as a function of time.b) Calculate the angular acceleration of the roller as a function of time.c) What is the maximum positive angular velocity?d) At what value of t does it occur?
A child is pushing a merry-go-round. The angle through which the merry-go-round has turned varies with time according to θ(t)= γt+ βt3, where γ = 0.406 rad/s and β = 1.30×10−2 rad/s3.a) Calculate the angular velocity of the merry-go-round as a function of time.b) What is the initial value of the angular velocity?c) Calculate the instantaneous value of the angular velocity ωz at t = 5.05 s .d) Calculate the average angular velocity ωav - z for the time interval t = 0 to t = 5.05 s .
Two children are riding on a merry-go-round. Child A is at a greater distance from the axis of rotation than child B. Which child has the larger tangential speed?A) Child BB) They have the same zero tangential speedC) Child AD) They have the same non-zero tangential speedE) There is not enough information given to answer the question.
a) Calculate the angular velocity of the second hand of a clock. State in rad/s.b) Calculate the angular velocity of the minute hand of a clock. State in rad/s.c) Calculate the angular velocity of the hour hand of a clock. State in rad/s.d) What is the angular acceleration in each case?
The angular acceleration of a wheel, as a function of time, is α = 5.0 t2 - 8.5 t, where α is in rad/s2 { m{rad/s^2}} and t is exttip{t}{t}in seconds. If the wheel starts from rest (θheta = 0, ω omega= 0, at t exttip{t}{t}= 0), determine a formula fora) the angular velocity ωomega as a function of timeb) the angular position θheta as a function of timec) evaluate ω at t = 4.0 sd) evaluate θ at t = 4.0 s