# How far does a ball thrown over a surface with friction slide until it starts rotating? [closed]

A bowling ball is thrown horizontally with initial velocity $v$ and no initial rotation over a surface with static coefficient of friction $\mu$. Determine the distance the ball travels ($d$) until it stops sliding and starts rotating.

I can't even start this problem (how do I know, mathematically, that the ball will stop sliding and will start rotating?).

Answer: $d=\frac{12v^2}{49\mu g}$

• The bowling ball decelerates due to friction, while at the same time this friction causes its angular velocity to increase. At some point in time the velocity of the point of the ball which touches the floor will be zero, at which point there is no kinetic friction anymore (I assume the coefficient of friction is for kinetic and not static friction). For this question you do have to assume that the bowling ball is of constant density, which is real life is not the case. May 31, 2015 at 14:31
• Welcome to Physics! Please note that Physics.StackExchange is not a homework help site. Please see this Meta post on asking homework-like questions and this Meta post for "check my work" problems. May 31, 2015 at 19:28

The coefficient of friction, $\mu$, will determine the torque on the ball as it slides. This torque is constant and can be defined as $\tau=\mu\,m\,g\,r$, where $m$ is the ball's mass, $r$ its radius and $g$ the acceleration due gravity.

So the ball's angular speed will increase according to $I\,\dot{\omega} =\tau$, where $I$ is the ball's mass moments of inertia. If you can work out $I$ and you can find the linear rate of increase of angular speed.

At the same time, the ball's speed $v$ will decrease following Newton's second law.

So now you can work out the speed $v(t)$ and angular speed $\omega(t)$ as a function of time $t$.

When the angular speed reaches the point where

$$\omega(t_0)\,r = v(t_0)\tag{1}$$

there is no relative slip between the ball and the surface. At this point, the friction force theoretically stops and the ball rolls onwards at a constant speed.

Your mission, should you choose to accept, is to solve (1) for $t_0$ and then you can work out the distance travelled.

• Can you explain why the equation (1) represents the exact point the ball starts rolling? I did solve the problem correctly, but it is not clear why it worked. May 31, 2015 at 14:15
• @ViníciusLopesSimões See the answers to this question. It's the state where the velocity of the ball's edge relative to the ball's center is the same as the velocity of the ground relative to the ball's center. Otherwise put, points at the edge momentarily stop as they make contact with the ground. May 31, 2015 at 22:06