A spherical bowling ball with mass m = 4 kg and radius R = 0.113 m is thrown dow

Question

A spherical bowling ball with mass m = 4 kg and radius R = 0.113 m is thrown down the lane with an initial translational speed of v = 8.6 m/s and no rotational speed. The coefficient of kinetic friction between the sliding ball and the ground is %u03BC = 0.3. Once the ball begins to roll without slipping it moves with a constant velocity down the lane.

1)

What is the magnitude of the angular acceleration of the bowling ball as it slides down the lane?

2)

What is magnitude of the linear acceleration of the bowling ball as it slides down the lane?

3)

How long does it take the bowling ball to begin rolling without slipping?

4)

How far does the bowling ball slide before it begins to roll without slipping?

5)

What is the magnitude of the final velocity?

6)

Explanation / Answer

The equations of motion for translation and rotation are: m dv/dt = -? m g I dw/dt = ? m g R The first is solved by v(t) = v0 - ? g t As for the second, using I = 2/5 m R^2 for a solid uniform sphere (ignoring the holes in the bowling ball), this is solved by w(t) = w0 + 5 ? g t / (2R) Therefore 1. angular acceleration is 5 ? g /(2 R) 2. linear acceleration is -? g 3. When is v(t) = w(t) R (pure rolling)? v0 - ? g t = w0 R + 5 ? g t / 2 (7 ? g /2) t = v0 + w0 R t =( 2 v0 + 2 w0 R)/( 7 ? g ) Assuming w0 = 0 (no spin is given in the throw) , this is t = 2 v0 / (7 ? g ) 4. s(t) = v(0) t + 1/2 a t^2 = 2 v0^2 / (7 ? g ) - 1/2 * (? g) ( 4 v0^2 / (7? g )^2 = 2/7 (v0^2 / (? g) ) - 2/49 (v0^2 / (? g) ) = 12 v0^2/( 49 ? g) Numerically this gives: 1. angular acceleration = 5*0.34*9.81 m/s^2 / ( 2 * 0.119 m) = 70.1 radians/second^2 2. linear acceleration = - 0.34 * 9.81 m/s^2 = -3.34 m/s^2 3. t = 2*8.9 m/s / ( 7 * 0.34 * 9.81 m/s^2) = 0.76 s 4. s = 12 * (8.9 m/s)^2 / (49 * 0.34 * 9.81 m/s^2) = 5.82 m

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