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Ch 12: Rotational KinematicsWorksheetSee all chapters
All Chapters
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

Concept #1: Rolling Motion (Free Wheels)

Example #1: Speeds at points on a wheel


Here we have a car that accelerates from rest for 10 seconds. The initial velocity of the car is zero and it takes 10 seconds accelerating. Its tires will experience 8 radians per second. V is the speed of the car. Let's draw a little car here. V is the speed of the car. This is a really crappy car. But w has to do with the wheel. If the car is moving that way, the wheel is spinning this way. I'm giving you the acceleration of the wheel, so I'm going to put it separately here because this is linear and I'm going to make a column here for angular. _ = 8. If the car is initially at rest, the w initial is also 0. The tires have a radius of 0.4 meters, so IÕm gonna write it down here that the radius of the tire is 0.4 meters. We want to know what is the angular speed of the tires after 10 seconds. After 10 seconds, what is w final for the tires? That's part A. This looks like a motion problem and it is. I got three motion variables here that are given and I'm asking for one. One of them is ignored. What's ignored here is the number of rotations. __ is my ignored variable sad face. This tells me that I should be using the first equation. w final = w initial + _t. w initial = 0, _ = 8*10, the answer is 80 rad/s. By the way, nothing new in this question. We've done stuff like this before. The part thatÕs new is Part B. Part B weÕre being asked for the speeds at the top, center and bottom of the tire. The tire is in rolling motion or you can think of it as the tire is a free axis or freewheel. This means that on top of the other equations we know, we're also going to be able to use the three equations that we just learned. Vtop will simply be 2Rw. 2, the radius is 0.4, w is 80. V center of mass in the middle is 1Rw, and V bottom is zero always. If you multiply V top, you get 64. The top is double what's in the middle so the middle must be 32 and the bottom is 0. That's it for this question. We have to find w final which is old stuff, then we have to find V top, Vcm and V bottom and we got them. Let me get over here so you can see the numbers. That's it for this one. Hopefully this makes sense. Let me know if you guys have any questions.

Practice: A long, light rope is wrapped around a cylinder of radius 40 cm, which is at rest on a flat surface, free to move. You pull horizontally on the rope, so it unwinds at the top of the cylinder, causing it to begin to roll without slipping. You keep pulling until the cylinder reaches 10 RPM. Calculate the speed of the rope at the instant the cylinder reaches 10 RPM.