1. Rotational Kinematics

Objectives: After completing this unit, you should be able to: Define and apply concepts of angular displacement, velocity, and acceleration. Draw analogies relating rotational motion parameters (, , ) to linear (x, v, a) and solve rotational problems. Write and apply relationships between linear and angular parameters.

Rotational Displacement,  Consider a disk that rotates from A to B: Angular displacement, q: B Measured in revolutions, degrees, or radians.  A 1 rev = 3600 = 2 rad The best measure for rotation of rigid bodies is the radian.

Definition of the Radian One radian is the angle  subtended at the center of a circle by an arc length s equal to the radius R of the circle. s 1 rad = 57.30

q = 40 rad And, 1 rev = 2p rad q = 6.37 rev Example 1: A rope is wrapped many times around a drum of radius 50 cm. How many revolutions of the drum are required to raise a bucket to a height of 20 m? q = 40 rad h = 20 m R And, 1 rev = 2p rad q = 6.37 rev

Example 2: A bicycle tire has a radius of 25 cm Example 2: A bicycle tire has a radius of 25 cm. If the wheel makes 400 rev, how far will the bike have traveled? q = 2513 rad s = q R = 2513 rad (0.25 m) s = 628 m

Angular Velocity Angular velocity, w, is the rate of change in angular displacement. (radians per second.) w = Angular velocity in rad/s. q t Angular velocity can also be given in terms of the frequency of revolution, f (rev/s or rpm): w = 2pf Angular frequency f (rev/s).

q = 50 rad q 50 rad t 5 s w = = w = 10.0 rad/s Example 3: A rope is wrapped many times around a drum of radius 20 cm. What is the angular velocity of the drum if it lifts the bucket to 10 m in 5 s? h = 10 m R q = 50 rad w = = q t 50 rad 5 s w = 10.0 rad/s

Example 4: In the previous example, what is the frequency of revolution for the drum? Recall that w = 10.0 rad/s. h = 10 m R Or, since 60 s = 1 min: f = 95.5 rpm

Angular Acceleration Angular acceleration is the rate of change in angular velocity. (Radians per sec per sec.) The angular acceleration can also be found from the change in frequency, as follows:

Example 5: The block is lifted from rest until the angular velocity of the drum is 16 rad/s after a time of 4 s. What is the average angular acceleration? h = 20 m R a = 4.00 rad/s2

Relating Angular and Linear Speed From the definition of angular displacement: s = q R and the definition of linear velocity: v = w R So, linear speed = angular speed x radius

Angular and Linear Acceleration: From the velocity relationship we have: v = wR and the definition of linear acceleration: ω ω a = aR a So, linear accel. = angular accel. x radius

Speed Example R1 A Consider flat rotating disk: B wo = 0; wf = 20 rad/s t = 4 s R1 = 20 cm R2 = 40 cm What is the final linear speed at points A and B? vAf = wAf R1 = (20 rad/s)(0.2 m); vAf = 4 m/s vBf = wBf R2 = (20 rad/s)(0.4 m); vBf = 8 m/s

Acceleration Example R1 A Consider flat rotating disk: B wo = 0; wf = 20 rad/s t = 4 s Consider flat rotating disk: R1 = 20 cm R2 = 40 cm What is the average angular and linear acceleration at B? a = 5.00 rad/s2 a = 2.00 m/s2 a = aR = (5 rad/s2)(0.4 m)

A Comparison: Linear vs. Angular

Select Equation: 0 - vo2 -(20 m/s)2 2s 2(100 m) a = -2.00 m/s2 a = = Linear Example: A car traveling initially at 20 m/s comes to a stop in a distance of 100 m. What was the acceleration of this car? Select Equation: 100 m vo = 20 m/s vf = 0 m/s a = = 0 - vo2 2s -(20 m/s)2 2(100 m) a = -2.00 m/s2

and 1 rev = 2π rad, so 50 rev = 314 rad Angular analogy: A disk (R = 50 cm), rotating at 600 rev/min comes to a stop after making 50 rev. What is the acceleration? wo = 600 rpm wf = 0 rpm q = 50 rev Select Equation: R and 1 rev = 2π rad, so 50 rev = 314 rad a = = 0 - wo2 2q - (62.8 rad/s)2 2 (314 rad) a = -6.29 rad/s2

Problem Solving Strategy: Draw and label sketch of problem. Indicate + direction of rotation (CW or CCW). List givens and state what is to be found. (q,wo,wf,a,t) Select equation containing the knowns and the unknown quantity that you are trying to determine, and solve for the unknown.

Note: net displacement is clockwise (-) Example 6: A drum is rotating clockwise initially at 100 rpm and undergoes a constant counterclockwise acceleration of 3 rad/s2 for 2 s. What is the angular displacement? R +a Given: wo = -100 rpm ? rad/s t = 2 s a = +3 rad/s2 q = -14.9 rad Note: net displacement is clockwise (-)

Bell Ringer A disk is rotating with an angular speed of ω. Where on the disk will an object have the greatest linear speed? Solution: v = ω r So, the further away from the center of the disk (r), the greater the linear speed (v).

Practice Question: Bonnie sits on the outer rim of a merry-go-round, and Klyde sits midway between the center and the rim. The merry-go- round makes one complete revolution every two seconds. Klyde’s angular velocity is: 1) same as Bonnie’s 2) twice Bonnie’s 3) half of Bonnie’s 4) 1/4 of Bonnie’s 5) four times Bonnie’s w Bonnie Klyde

Bonnie and Klyde Solution Bonnie sits on the outer rim of a merry-go- round, and Klyde sits midway between the center and the rim. The merry-go-round makes one complete revolution every two seconds. Klyde’s angular velocity is: 1) same as Bonnie’s 2) twice Bonnie’s 3) half of Bonnie’s 4) 1/4 of Bonnie’s 5) four times Bonnie’s The angular velocity of any point on a solid object rotating about a fixed axis is the same. Both Bonnie & Klyde go around one revolution (2p radians) every two seconds. w Bonnie Klyde