Chapter 8

Kinematics of Rotation

Radian Measures: The arc-length divided by the radial distance is defined as a radian.

Movement of rigid bodies can be described in terms of rotation and translation.

	Rectilinear (along a straight line) translation
	Curvilinear (along an arc) translation
	Rotation (about a point within the body)
	Rotation (about a point outside the body)
	Rotation and translation

Angular Displacement: When a rigid body rotates about a fixed axis, the angular displacement is the angle swept out by a line passing through any point on the body and intersecting the axis of rotation perpendicularly. By convention, the angular displacement is positive if its counterclockwise and negative if it is clockwise.

Example 8-1

A particular hawks eye can just distinguish objects that subtend an angle no smaller than about . (a) How many degrees is this? (b) How small an object can the bird distinguish when flying at a height of 100 m?

Average Angular Velocity:

Example 8-2

A gymnast on a high bar swings through two revolutions in a time of 1.90 s. Find the average angular velocity of the gymnast.

Instantaneous angular velocity:

Example 8-3

(a) An old phonograph record clockwise at . What is its angular velocity in rad/s? (b) If a CD rotates at 22.0 rad/s, what is its angular speed in rpm? (c) Find the period of a record that is rotating are 45 rpm.

Angular Acceleration:

Tangential Acceleration:

Example 8-4

If the angular velocity of the pulley in the figure is -8.4 rad/s at a given time, and its angular acceleration is -2.8 , what is the angular velocity of the pulley 1.5 s later?

From these equations we can derive the following equations.

Equations of Constant Angular Acceleration

Example 8-5

The disk in the figure is rotating about its central axis like a merry-go-round. The angular position of a reference line on the disk is given by

,

with t in seconds, in radian, and the zero angular position as indicated in the figure.
(a) Graph the angular position of the disk versus time from t = -3.0 s to t = 6.0 s. Sketch the disk and its angular position reference line at t = -2.0 s, 0 s, and 4.0 s, and when the curve crosses the t axis.
(b) At what time does reach the minimum value shown in the graph from (a)? What is that minimum value?
(c) Graph the angular velocity of the disk versus time from t = -3.0 s to t = 6.0 s. Sketch the disk and indicate the direction of turning and the sign of at t = -2.0 s and 4.0 s, and also at .
(d) Use the results in parts (a) through (c) to describe the motion of the disk from t = -3.0 s to t = 6.0 s.

Example 8-6

A grindstone rotates at constant angular acceleration . At time t = 0, it has an angular velocity of and a reference line on it is horizontal, at the angular position .
(a) At what time after t = 0 is the reference line at the angular position = 5.0 rev?
(b) Describe the grindstone's rotation between t = 0 and t = 32 s.
(c) At what time t does the grindstone momentarily stop?

Example 8-7

While operating a carnival ride called the Rotor (a rotating cylindrical ride), you spot a passenger in acute distress and decrease the angular speed of the cylinder from 3.40 rad/s to 2.00 rad/s in 20.0 rev, at constant angular acceleration.
(a) What is the constant angular acceleration during this decrease in angular speed?
(b) How much time did the speed decrease take?

Example 8-8

A cyclist traveling at 5.0 m/s uniformly accelerates up to 10.0 m/s in 2.0 s. Each tire of the bike has a 35 cm radius, and a small pebble is caught in the tread of one of them. (a) What is the angular acceleration of the pebble during those two seconds? (b) Through what angle does the pebble revolve? (c) How far around the wheel does the pebble travel during that accelerating interval?

Example 8-9

In a microhematocrit centrifuge, small samples of blood are placed in heparinized capillary tubes (heparin is an anticoagulant). The tubes are rotated at 11500 rpm, with the bottom of the tubes 9.07 cm from the axis of rotation. (a) Find the linear speed of the tubes. (b) What is the centripetal acceleration at the bottom of the tubes?

Example 8-10

Suppose the centrifuge in Example 8-9 is just starting up, and that it has an angular speed of 8.00 rad/s and an angular acceleration of . (a) What is the magnitude of the centripetal, tangential, and total acceleration of the bottom of a tube? (b) What angle does the total acceleration make with the direction of motion?

On to Part 2