# ROTATION I

I. Rotation usually takes place about a fixed axis

A. If a radius of length r rotates about a fixed axis at O from position 1 to position 2, the angular position is denoted by the Greek letter theta, q . Theta is usually measured in radians and is the length of the arc swept out by the radius divided by the length of the radius. A change in angular position is an angular displacement, D q .

II. If a radius vector rotates with time, the change in angle divided by the time interval is the average angular speed, denoted by the greek letter omega, w .

A. What is the instantaneous angular speed?

III. If the angular speed changes with time, then we have an angular acceleration, denoted by the Greek letter alpha, a .

A. What is the average angular acceleration?

B. What is the instantaneous angular acceleration?

IV. The direction of the angular velocity is determined by a right-hand convention.

V. We can develop equations of motion for a constant angular acceleration.

VI. Relate angular quantities to linear quantities

A. How is the angular quantity q related to the linear displacement s?

B. How is the angular speed w related to the linear speed v?

C. How is the angular acceleration a related to the linear acceleration a?

D. How do we express centripetal acceleration in terms of the angular variables?

Example

An old phonograph record rotates at 45 rpm. A bug lands on it so that she is 10 cm from the axis of rotation. Find

a) its angular speed

b) the linear speed of the bug

c) the centripetal, angular, and linear accelerations

d) the angular speed and the tangential speed of a second bug found at r = 5.0 cm

Suppose that in the previous example the bug at r = 10 cm landed on the record when it was not rotating. It starts to accelerate uniformly and after 2.5 seconds its instantaneous angular speed is 18 rad/s.

a) What is the average angular acceleration?

b) Determine the tangential acceleration, the centripetal acceleration, and the total acceleration at t = 2.5 s.