What Happens When a Slipping Sphere Begins to Roll Without Slipping?

[jbphys303]

Homework Statement

A uniform density sphere is released such that it has an angular speed of 10 rev/sec and no initial linear velocity. The angular velocity vector is perfectly perpendicular to the linear momentum vector. Initially the ball slips as it moves along the surface, but after time t pure rolling without slipping begins.

The coefficient of friction between the sphere and the surface is 0.21.
The radius of the ball is 0.08m
mass sphere= 7.3 kg
w= 10 rev/sec

#1: How fast is the ball rolling at time t?

#2: How long did it take to reach this speed?

#3: How much energy was lost between time t=0 and t=t?

Homework Equations

Rotational Inertia (I)=2/5mr^2
kinetic energy (rotational)= 1/2Iw^2
ke= 1/2mv^2
v=rw

The Attempt at a Solution

I think I can figure it out once I know how much energy is lost by friction
So far I set up the following equation
1/2Iw^2-(energy lost by friction)= 1/2Iw(final)^2+ 1/2mv^2
Any help would be appreciated

 

[ehild]

The motion of the ball is composed of translation of its CM and of rotation around the CM. The kinetic friction points forward, accelerates translation and its torque with respect to the CM decelerates rotation. At the time when v=rw, static friction takes place and the ball rolls.

Use both equation for linear acceleration ma = F and for angular acceleration I*dw/dt=-RF. Solve for v and w. Find the time when wR=v.

ehild