Rolling Objects With Slipping Question

[Axxaaa]

Consider a ball of radius r and total mass m, with a nonuniform but radially symmetric mass distribution inside it, so that it can have an almost arbitrary moment of inertia I. (a) If this ball is projected across a floor with an initial velocity v, and is at first purely sliding across the floor, what is its final velocity when it is rolling without slipping. (b) What is the total energy of the ball?

Equations:
T = I * α
v = r * w
a = r *α
rotational-kinematics formulas

I do not know how to even begin. For the ball to start rolling without slipping, v must equal rw, which means rw must increase. There is no mention of force of friction that might provide the angular acceleration in this problem, so I am quite lost.

 

[tiny-tim]

Hi Axxaaa! Welcome to PF!

There is no mention of force of friction that might provide the angular acceleration in this problem, so I am quite lost.

Call the kinetic friction "F", and write out the equations for a and α …

you should find that F cancels out in the end.

Show us what you get. ​

 

[haruspex]

Another approach is to consider the angular momentum about a point on the floor. Since friction acts along the floor it will not affect that.