## motion involving both translation and rotation

Hello everyone. Thanks for all the help today. I know I have asked a lot of questions. This is my last one today.

A spherical billard ball is sliding with a speed v_o. It has a radius R, mass M, and there is a friction coefficient mu. Determine the distance the ball travels before it begins to roll smoothly on the surface.

The correct answer is supposed to be 12v_o^2/(49*mu*g).

I know that the kinetic energy K = (1/2)MV_cm^2 (1 + B)

K = (1/2)MV_cm^2 ( 1 + 2/5) - mu*Mg*x
K = (1/2)Mv_cm^2 ( 7/5) -mu*Mg*x

K = (7/10)Mv_cm^2 -mu*Mg*x
x = (7/10)Mv_cm^2 /(mu*mg)

However, this is not the right answer. Where did I go wrong? Is it possible to solve this with energy concept? I don't know how to do it with Newton's Law. I don't know how to get velocity out of the problem.
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 Recognitions: Homework Help Science Advisor It would probably be a bit messy to do this with just work/energy. The work done by friction as the ball slides on the table is not the force times the distance the ball moves. If the ball is sliding without rolling, the work done is friction times the distance the center of the ball moves. If the ball is rolling, the work done by friction is zero. As the ball begins to rotate while still sliding, the work done per unit distance the ball moves gradually diminishes. It is easier to look at the force of friction causing two things to happen simultaneously. First, it declelerates the center of mass of the ball by Newton's 2nd law: F = ma. Second, it applies a torque to the ball that causes it to rotate: torque = Fr = Iα. Using the connection between v and ω when the ball is rolling you can determine when slipping stops and pure rolling begins
 I don't really understand what do. From what you told me, The net force in the horizontal direction is ma, with friction being mu*mg The net torque is I*(alpha), and I know that I for a sphere is (2/5)MR^2 alpha is a/R so I = (2/5)*MA*R and F = mu*Mg The answer has a 12 and 49, but I don't know where they're coming from. I know d = (1/2)at^2, but I don't know what to do with it.

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