STARTER

Consider two points, A and B, on a spinning disc.

1. Which point goes through the greatest distance in 1 revolution?

2. Which point goes through the most degrees in 1 revolution?

3. Which point has the greatest velocity?

4. Which point has the greatest angular speed? ( r.p.m.’s)

Practice: Circular Motion

RADIAN MEASURE q = 1 radian when s = r

s = rq This is true

when q is measured in radians.

180 degrees = pradians

1 radian = 57.3 degrees

1 revolution = 2p radians

Example:

Convert the following:

1. 45 degrees = ________radians

2. 1.5 radians = _________ degrees

3. .8 revolutions = ________ radians

1. 45 degrees (p rad / 180 degrees) = .785 radians

2. 1.5 radians ( 180 degrees / p radians ) = 85.9 degrees

3. .8rev ( 2 p rad / rev) = 5.03 rads

Angular Velocity wAngular velocity is how many radians per second an object moves through.

w = Dq / Dt(rad/s)

Example: A disc spins through 2 revolutions in 3 seconds. What is the angular velocity of the disc in radians/second?

Solution : = Dq 2 rev ( 2p rads / 1 rev) = 4p rads = 12.6 rads

so w = Dq / Dt = 12.6rads/3 sec = 4.12 rads/sec

Tangential Velocity v

A point on a disc rotating with an angular velocity w, has a tangential velocity in m/s.The velocity of the point depends on how far it is from the center, in fact:

v = wrExample: A disc spins through 2 revolutions in

3 seconds. What is the velocity of a point 10cm from the disc’s center? Solution : v = wr = (4.12 rads/sec )(.10m) = .412 m/s

Angular ( )w and Tangential Velocity (v)

For a rotating object, all points have the same w, but different tangential velocities.

v = wr

Angular Acceleration a

If the angular speed changes with time, there will be an angular acceleration, a.

= /a Dw Dt (rad/s2 )

Example: A disc spinning at 10rad/s, slows to 5 rad/s in 2 seconds. What is the angular acceleration?

Solution : a = /Dw Dt = ( 5 - 10) / 2 = -2.50 rad/s2

EXIT

Two children are on a rotating carnival ride. Write a short paragraph comparing the angular velocity and the tangential velocities of each child.

STARTER

If the chain moves at 1 m/s and the radius of the rear gearis 8cm, what is the angular speed of the rear gear in rad/s ?

w = v/r = 1/.08 = 12.5 rad/s

Centripetal Acceleration ac

If an object is moving in a circle, it has an acceleration that points to the center of the circle, called the centripetal acceleration, ac.

ac = v2/r= w2rExample: A disc spins at 12 rad/s. What is the centripetal acceleration of a point 10cm from the disc’s center? Solution : a = w2r = (12)2(.10) = 14.4 m/s2

Summary

w = Dq / Dtv = wr

= /a Dw Dt

ac = v2/r= w2r

Angular Velocity

Tangential Velocity

Angular acceleration

Centripetal Acceleration

All The Vectors for Rotation

Tangential Velocity ( always there )

Centripetal Acceleration ( always there)

Tangential Acceleration ( only there if its speeding up or slowing down)

Total Acceleration ( the vector sum of at and ac )

Kinematic Equations for Constant Angular Acceleration

1. wf = wi + at

2. Dq = wit + (1/2) at2

Example

A motor starts from rest and accelerates to 40 rad/s in 10 seconds.

1. What is the angular acceleration?2. How many radians does the motor turn through?

To get a, you need an equation with a in it, but without qf. Which

one is it?

1. wf = wi + at

40 = 0 + 10a or

a = 4.00 rad/s2

To get Dq use 2.

Dq = 0 + (1/2)(4)(42) = 32 radians

2. Dq = wit + (1/2) at2

Application: Circular Motion Problem Set

Connection

An audio CD head reads the information from thedisc at a constant rate. This means that thetangential velocity of the disc where the read head is must be constant.

This means that as the read head moves closer to the center of the disc, v = wr = constant.

So as r gets smaller, what must happen to w?Explain.

EXIT :4 Minute Writing

Carefully describe the difference between angular velocity and tangential velocity.

Then consider two different points on a spinning fan blade ( point A is closer to the center of the blade and point B is near the outer edge).

Compare their tangential and angular velocities.