What have I done wrong? (torque and angular momentum)

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

A billiardball is hit from rest by the cue at height "h" from the table with a force F in the time interval Δt. The mass M and radius R of the ball is known as well as the moment of inertia which is [itex]I=\frac{2}{5}MR^2[/itex]
Find an expression for the height "h" at which the billiardball will roll without slipping when it is hit.

2. Relevant equations
Condition for roll with no slipping:

3. The attempt at a solution
I start by finding an expression for [itex]v_{cm}[/itex].

Center of mass differentiated by Δt gives:

Newton’s 2nd law:

Isolating velocity gives:

Since the velocity is equal to the velocity of the center of mass:

Now I find an expression for the angular velocity.

The net torque is given by:

The only force is the force F from the cue which gives the torque [itex]τ=F(h-R)[/itex] where (h-R) is the perpendicular length from the force F to the center of mass of the ball.

The angular velocity is:

Now I insert the velocity of CM and the angular velocity into the rolling without slip equation:

And I end up with:

But my expression for the height has the mass squared in it. What did I do wrong?


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