Bowling Ball Changing Rotational Direction

[Curtiss Oakley]

Homework Statement

A uniform bowling ball of radius R, mass M is thrown down a horizontal lane with initial horizontal speed v0 and backspin(as shown below) with initial angular speed ω0, such that v0 > Rω0. So after the ball makes contact with the horizontal lane, it rolls with slipping on the lane. The kinetic frictional force fk acting on the ball causes a center of mass acceleration of the ball acom to slow down its linear motion. Meanwhile this kinetic frictional force fk also produces a torque that causes an angular acceleration of the ball to speed up its angular motion. When speed vcom has decreased enough and the angular speed ω has increased enough, the ball will start to roll without slipping. The coefficient of kinetic friction between the ball and lane surfaces is μk. The moment of inertia of the ball about its center of mass is Icom=2/5MR2.
(a) Find the center of mass velocity v and angular velocity ω of the ball as a function of time after it makes contact with the lane and up to the point when it starts to roll without slipping. Note: directions are important here!
(b) Find the center of mass velocity vf and angular velocity ωf of the ball when it just starts to roll without slipping.
(c) Find the frictional force between the bowling ball and horizontal surface during the ball’s rolling without slipping phase;
(d) Find the center of mass acceleration acom of the ball during its rolling without slipping phase; (e) Find the angular acceleration α of the ball during its rolling without slipping phase;
(f) If the thrower of the ball wants the ball to come to rest before it can reach the point that it will start to roll without slipping, how must the initial conditions, v0 and ω0, be related?

Relevant Equations

Torque net=I(alpha)
Fnet=Ma
v=at+v0
v=r(omega)

I already have a and b, but want to see if anyone is willing to verify my answer for part c. I get 0 for the frictional force between the ground and ball, which would lead d and e to be 0 as well. Physics is rarely that easy so I wanted to make sure I didn’t miss anything.

 

[kuruman]

It's not a matter of physics being easy for part (c). It's a matter of reasoning things out. If there were no friction, would the ball start rolling without slipping or would it keep sliding forever?

 

[Curtiss Oakley]

It's not a matter of physics being easy for part (c). It's a matter of reasoning things out. If there were no friction would the ball start rolling without slipping or would it keep sliding forever?

Friction is what causes the ball to start rolling without slipping. If there wasn’t any it would roll forever. But after it starts rolling without slipping, that’s the period I’m concerned about.

 

[kuruman]

OK, let's look at this from a different angle. Suppose the ball has reached the stage of rolling without slipping. Neglecting dissipative forces, which seems to be the case here, the ball will keep rolling forever at constant velocity.
What does this imply about the force of static friction? I think you know the answer.
Now suppose the ball hits a frictionless patch on the surface. How would the ball's motion change and why?

 

[Curtiss Oakley]

OK, let's look at this from a different angle. Suppose the ball has reached the stage of rolling without slipping. Neglecting dissipative forces, which seems to be the case here, the ball will keep rolling forever at constant velocity.
What does this imply about the force of static friction? I think you know the answer.
Now suppose the ball hits a frictionless patch on the surface. How would the ball's motion change and why?

Velocity would be constant on a frictionless patch. I don’t see how it could change without any forces acting on it.

 

[kuruman]

Velocity would be constant on a frictionless patch. I don’t see how it could change without any forces acting on it.

That's right, but would the ball still rotate about its center or would it just slide without rotating?