# Rolling motion of an accelerated body

I recently read in a book that static friction is zero for a rolling body, and this explains why it doesn't slow down the rolling body. However, if static friction is zero, how does the body grip the floor to prevent it from slipping?

In the book I read, in the free body diagram of a rolling body moving with uniform velocity, static friction was not included. But in the free body diagram of a rolling body upon which a horizontal force has been applied, the force of static friction has been included and friction has also been included in the equations of motion of the rolling body.

Why has friction been included only when an external force is applied?

And is static friction really zero for a rolling body?

• What book? Didn't the book mention anything about "rolling friction" as well? Commented Dec 2, 2013 at 12:21
• I know about rolling friction. But I'm confused about static friction Commented Dec 2, 2013 at 13:11
• I am unclear on what you are asking the Physics community for? This question is specific to the book mentioned. Commented Dec 2, 2013 at 13:56
• Just forget the book for the time being, my question is does static friction cause an acceleration only when an external force is applied on the rolling body to counter the torque applied? Commented Dec 2, 2013 at 15:12
• Related question by OP: physics.stackexchange.com/q/88587/2451 Commented Dec 2, 2013 at 15:32

If you could suddenly reduce gravity to zero, the wheel would continue rolling at the same angular velocity and the center would continue with he same linear velocity. Hence the is no friction force at the contact point. The only force acting is the vertical force that balances gravity. The wheel moves under conditions of constant angular and linear momentum.

When one talks about static friction one generally thinks of the coefficient that enters in the static friction force $(\mu_{static})$. The force that you can apply to a body without moving it is proportional to this coefficient $||{\vec{F}}||\propto \mu_{static}$.

You should not confuse this friction with the friction that occurs for a body that moves $(\mu_{dynamic})$ or roll $(\mu_{roll})$.

In general (in most cases) $\mu_{static}> \mu_{dynamic} > \mu_{roll}$.

I recently read in a book that static friction is zero for a rolling body, and this explains why it doesn't slow down the rolling body. However, if static friction is zero, how does the body grip the floor to prevent it from slipping?

It does not "grip" in such idealization. The angular velocity and the velocity of the center of mass are just "fine tuned" so that the contact point does not slide on the floor.

Why has friction been included only when an external force is applied?

When the external force is applied horizontally on the rolling body, if there was no floor and no gravity, the translational and rotating motion would accelerate in such way that would not prevent the fine-tuned state of the no-slip condition - the lowest point would slip on the imaginary floor.

In reality, the floor and gravity are present, so as soon as the external force begins to act, the floor starts to act with static friction force on the body with such magnitude and direction that prevents further slip (the actual initial slip is so small that it is unobservable). This will happen if the external force is not too strong; in the case it is too strong, the rolling body will slip discernibly.

Yes the static friction of the body is zero if it is rolling. This is because, the velocity of the lowermost point (i.e. the point of contact is zero by definition of pure rolling). So, as there is no tentative motion of the lowermost point, the static friction is $0$.

But whenever a force is applied, the lowermost point has a tendency to slide. As a result a frictional force has to act is the opposite direction in order to maintain the zero velocity of the lowermost point otherwise it won't remain a pure rolling motion. However this friction is not static friction, it is rolling friction.

Think about a vehicle. Whenever it moves, a force is applied on the wheels. This tends to rotate them because of a couple produced by this force and the rolling friction.