A bowling ball of mass $M$ and radius $r_0$ is thrown along a level surface so that initially ($t = 0$) it slides with a linear speed $v_0$ but does not rotate. As it slides, it begins to spin, and eventually rolls without slipping. How long does it take to begin rolling without slipping?
I am confused with the textbook solution. I understand that kinetic friction first acts on the ball. The linear velocity is given by $$ V_{CM} = v_0 – \mu g t$$
The angular acceleration is given by $$I \alpha = \Sigma \tau \implies \alpha = 5 \mu_k g/2r_0 $$
The book states this angular acceleration is constant (presumably from $t=0$). How can one arrive at this conclusion?
I'm confused how this can this be true if the force causing the torque changes from kinetic friction to static friction (when the ball starts to roll without slipping)?
See the full book solution here.
Source: Giancoli's Physics for Scientists and Engineers.
Best Answer
I think that your skepticism comes about because you intuitively think that the force of kinetic friction should change gradually to static friction as the ball speeds up, since the relative motion between the ball's spinning surface and the ground decreases to zero. Your textbook assumes that this transition is actually instantaneous, and that the kinetic friction force is exactly the same until there is no relative motion at all, at which point the friction is entirely static. It seems counter-intuitive, but that's actually how it is.
If you've ever seen someone play a cello or any stringed instrument with a bow, you can see a little better how this works. A cellist steadily moves his bow across the string, which vibrates as it slips and catches on the sticky bow, switching between kinetic and static friction. If friction had a gradual transition, I imagine that the bow would just push the string to a certain displacement and reach equilibrium, and we'd be robbed of a great instrument, and a lot of excellent music.
Does your problem make more sense now? The angular acceleration and the torque are constant, because the kinetic friction force is constant until it disappears. Once the ball spins with the floor, there's no torque on the ball because it's just rolling—to the floor it seems to be still. But as the ball is spinning up (and the slipping is slowing down), the force on it remains the same. Friction is just weird like that.