Analysis of Billiards Ball Motion Following a Horizontal Queue Impact

[besnik93]

1. Encountered with cue to a massive billiards ball, which is initially at rest, see figure uploaded. The ball has radius R and mass M. The queue hits with force F horizontally into the bale height h above the table, and the shock lasts a very short time Δt.
It is reported that the moment of inertia of the ball with respect to its center of mass is
I = 2/5 * M * R^2

The movement after the shock is a combination of a translational movement and a rotation about an axis through the center of gravity perpendicular to the plane of the paper.

a) Determine the speed of the billiard ball's center of mass and billiard ball's angular momentum with respect to the center of mass immediately after the collision. The answers must be expressed by the known sizes M, h, R, F and Δt.

b) At what height should the queue hit the ball to the ball immediately after the collision rolls without slipping?

The Attempt at a Solution

a) i think of focusing on the the center of mass, but how, i don't know..

b) i know that i need to focus on the expression of the mass center point of the speed and bale angular velocity. But i can't move on.

so i hope someone can help me, please..

 

[rcgldr]

Note that the change in linear momentum = F x Δt, called an impulse. The change in angular momentum would equal T x Δt, where T equals torque. Can you express torque as an equation using F, h, and R?

 

[besnik93]

I don't know how to express that to make sense

 

[paisiello2]

You guessed right with using the center of mass as the reference point.

The speed of the ball after the perfectly elastic collision is a very simple conservation of momentum:

F Δt = M v

The speed of the ball's angular momentum would then be conservation of angular momentum where you take the moment or torque about the ball's center of mass:

F Δt (h-R) = I ω

I think question b) is not stated quite correctly. I think it should read as follows:

b) At what height should the queue hit the ball so that the ball immediately after the collision rolls without slipping?

 

[besnik93]

b) yes that's it