Bowling Bowl - Center of mass - Friction - Angular Acceleration/Velocity

[greenarrow74]

1. The problem statement, all variables and given/known data

A bowling ball with Mass M and Radius R is thrown down an alley so that the balls center of mass has an initial horizontal velocity of Vo and the ball is not rotating. Kinetic friction force is f^k. Find:

a) an expression for the acceleration and velocity of the ball's center of mass (a) as a function of time (t)
b) in terms of M, R, Vo, and f^k, write down an expression for the angular acceleration and angular velocity as a function of time
c) at some time t^r, the ball will stop slipping on the alley and will start to roll w/out slipping. What are the conditions for this to happen
d) using these conditions in part (c), find the time t^r at which the ball starts to roll w/out slipping (in terms of M, R, Vo and f^k


3. The attempt at a solution

a) a=Vo/dt ?
b) angular acc = Torque/I
Torque = R x F
F= M X A
I = 2/3 x M x R^2 ?
c) v= R x w (ang velocity)
d) ???

Thanks so much for the help!!

[Doc Al]

3. The attempt at a solution

a) a=Vo/dt ?
Use Newton's 2nd law to find the acceleration, then apply standard kinematic formulas.
b) angular acc = Torque/I
Torque = R x F
F= M X A
OK. Do the same thing as in part a, only for rotation.
I = 2/3 x M x R^2 ?
That's for a spherical shell--a bowling ball is solid.
c) v= R x w (ang velocity)
Good--that's the condition for rolling without slipping.

The basic idea is this. The friction force does two things: It slows the translational speed as it raises the angular speed. Solve for the point at which those speeds meet the condition for rolling without slipping.