3/11/16

Rotational Motion Angular Position

In physics we distinguish two types of motion for objects:

• Translational Motion (change of location): • We will measure Whole object moves through space. angular position in radians: • Rotational Motion - object turns around an axis (axle); axis does not move. (Wheels) • Counterclockwise (CCW): positive We can also describe objects that do both at rotation the same time! • Clockwise (CW): negative rotation

Linear Distance d vs. Angular Simple vs. Complex Objects distance Δθ

Model motion with just Model motion with For a point at Position position and Rotation radius R on the wheel, R d = RΔθ

ONLY works if Δθ is in radians!

Speed in Rotational Motion Analogues Between Linear and Rotational Motion (see pg. 303) • Rotational Speed ω: rad per second

• Tangential speed vt: distance per second • Two objects can have the same rotational speed, but different tangential speeds!

1 3/11/16

Relationship Between Angular and Linear Example: Gears Quantities

• Displacements • Every point on the • Two wheels are rotating object has the connected by a chain s = θr same angular motion • Speeds that doesn’t slip. • Every point on the v r • Which wheel (if either) t = ω rotating object does • Accelerations not have the same has the higher linear motion rotational speed? a = αr t • Which wheel (if either) has the higher tangential speed for a point on its rim?

Angular Acceleration Directions Corgi on a Carousel

• If the angular acceleration and the angular • What is the carousel’s angular speed? velocity are in the same direction, the • What is the corgi’s angular speed? angular speed will increase with time. (Relative to ground or carousel)? • If the angular acceleration and the angular • If the corgi’s owner spins up the carousel velocity are in opposite directions, the in a time span of 0.5 s, what is the angular angular speed will decrease with time. acceleration of the carousel? • Estimate the corgi’s tangential speed (hint: how big do you think the carousel is?)

Example: CD drives Angular Acceleration In a CD player, the CD spins up from rest to about • Change in angular 500 rpm (revolutions per minute) in a span of velocity -> angular about a half a second. acceleration! A. What is the angular acceleration of the CD while it is spinning up? • However, even if angular velocity is B. What is the change in angle (Δθ) that the CD constant, each point travels through during this time? also has centripetal C. Once the CD is up to 500 rpm, what is the acceleration (due to tangential speed of a point on the outer edge of change in direction of the disc? What about at the inner edge of the v ) disc? t

2 3/11/16

Centripetal Acceleration and Circular Motion & Acceleration Angular Velocity • An object moving in uniform circular • The angular velocity and the linear velocity motion (ω constant) is still accelerating! are related (v = ωr) • Acceleration = changes in speed OR • The centripetal acceleration can also be changes in direction ! Recall from CH 6 related to the angular velocity • a = v 2/R = ω2R, pointed towards center of c t a 2r circle C = ω

3