Sliding time before a ball begins to roll on a horizontal surface

[NZBRU]

Homework Statement

A ball is released along a horizontal surface with a co-efficient of friction Us at a speed V. Find the time it takes for the ball to start rolling (leave the moment of inertia as I).

Homework Equations

Quite a few, all of them are stated in my working (that I believe are relevant).

The Attempt at a Solution

Spoiler

 

[Simon Bridge]

What is your reasoning?

 

[NZBRU]

After reviewing my lecture notes I found that I copied down a different solution. I attempted to redo the problem but came up with this solution, I must of either copied the answer incorrectly or made a mistake in the above working.

In regards to reasoning The Torque net must be equal to the frictional force acting directly below the ball multiplied by the radius, R. I assumed that the initial rotational velocity was zero. Using Torque(net) = I x alpha I found an expression for alpha. As the initial rotational velocity was equal to zero alpha = w/t.

I made the assumption that then the ball stopped rolling, V=Rw and solved for V. Using the kinematic equation v = u + at I solved for u (the initial linear velocity) and then solved for t.

 

[ehild]

Your method and result are both correct. Be careful with the signs, sometimes you wrote them incorrectly. The friction accelerates rotation but decelerates translation: Initially the ball slides, so α=dω/dt=μmg/I, ω=μmgt/I. μ is the coefficient of kinetic friction. At the same time, friction decelerates the translational motion of the CM: a=-μg, v=v₀-μgt.

ehild