What's the distance this ball travels?

Question

A billiard ball of radius a is initially spinning about a horizontal axis with angular speed w_0 and with zero forward speed. If the coefficient of sliding friction between the ball and the billiard table is µ_k, find the distance the ball travels before slipping ceases to occur.

Answer 1

At time zero, consider a FBD of the ball in the translational frame of reference. There is a force at the bottom of the ball in the horizontal direction equal to m*g*µ_k, where m is the mass of the ball.

Therefore there is an acceleration of the ball in the horizontal of g*µ_k

since there is no starting forward speed, the equation of velocity of the ball is
v(t)=g*µ_k*t

Now consider the rotational frame of the ball at the horizontal axis. There is an initial angular speed of w_0 and a torque of m*g*µ_k*a. The Torque=I*alpha, where I=2/5 *m^a^2
so alpha =g*µ_k*5/(a*2)
Using this the equation for angular speed is
w(t)=w_0-g*µ_k*5*t/(a*2)

When slipping stops the w(t)*a=v(t)
or
w_0*a-g*µ_k*5*t/2=g*µ_k*t
solve for t
w_0*a=g*µ_k*t*(5/2+1)
w_0*a=g*µ_k*t*7/2
t=w_0*a*2/(g*µ_k*7)

now that we know t, compute the distance of travel using
s(t)=.5*g*µ_k*t^2

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