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ajcoelho
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Homework Statement
This problem has already been discussed in lots of places through internet but none of them seems to be the correct answer in my opinion. So i tried to solve it and i'd like you to check my reasoning.
So the problem is:
If a billiard ball is hit in just the right way by a cue stick, the ball will roll without slipping immediately after losing contact with the stick. Consider a billiard ball (radius r , mass M) at rest on a horizontal pool table. A cue stick exerts a constant horizontal force F on the ball for a time t at a point that is a height h above the table's surface (see the figure). Assume that the coefficient of kinetic friction between the ball and table is [itex]\mu[/itex]
Determine the value for h so that the ball will roll without slipping immediately after losing contact with the stick.
Express your answer in terms of the variables r, F, [itex]\mu[/itex], and appropriate constants
Homework Equations
So I atched the problem only during the time F is acting! And during this time we have two forces causing torque: Fa and F!
So [itex]\Sigma\tau[/itex] = [itex]\mu[/itex]mgr+F(h-r)
We also know that [itex]\tau[/itex] = I [itex]\alpha[/itex] = 2/5 mr^2 [itex]\alpha[/itex]
Solving for [itex]\alpha[/itex] we get 5[F(h-r)+[itex]\mu[/itex]mgr]/2mr^2
To determine the velocity caused by the impulse (Ft) we have:
[itex]\Delta[/itex]p = Ft (where p is linear momentum)
This is equal to: m(v0 - vf) = Ft
Solving for vf we get Ft/m
The Attempt at a Solution
Now, to have pure rolling imediatly after the stick lose contact with the ball, at the instant t, [itex]\omega[/itex]= vf/r
Cause we already know vf, it gets: [itex]\omega[/itex]= Ft/mr
Now comes the part that I have doubts...:
From equation [itex]\omega[/itex]= w0 + [itex]\alpha[/itex]t
and saying that w0=0
then [itex]\alpha[/itex]=[itex]\omega[/itex]/t
Cause we know [itex]\omega[/itex] and [itex]\alpha[/itex], solving for h we get:
h = r[itex]\frac{7F-5umg}{5F}[/itex]
I'm not sure if this is the correct solution so I ask you guys to help me with this.
Thanks a lot!
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