# How does a cilinder that rolls without slipping lose energy? [duplicate]

Suppose a cilinder that rolls without slipping on a horizontal rough surface. The slipping is prevented by friction, which therefore slows it down, so it loses kinetic energy. But no dissipative work is done. How is this possible?

## marked as duplicate by Qmechanic♦Aug 9 '18 at 2:49

• An object can roll without slipping without friction acting on it. – Aaron Stevens Aug 8 '18 at 23:57
• But slet us suppose there is friction – thedude Aug 8 '18 at 23:58
• There is a small amount of air resistance, and there is also a very small amount of rolling resistance. – David White Aug 8 '18 at 23:58
• I want to consider a situation in which friction is the only force present. Or, if this is impossible, I want ro understand why – thedude Aug 9 '18 at 0:17
• The question arises because in pure rolling-without-slipping there’s no relative motion between the surface and cylinder, hence (in that ideal case) no work done. – Bob Jacobsen Aug 9 '18 at 0:40

The issue here is that you are assuming if something is rolling without slipping it must be that friction is acting on the body. This is just not the case in the ideal situation without rolling friction.

If you have an object rolling without slipping with no other torques acting upon the body, then friction is not acting on the object and it continues rolling with the same linear and angular velocities. It is similar to a book sitting on a table. No horizontal net force is acting on this book, so there is no static friction force acting on it.

Now let's say we have our rolling object and we try to apply an external torque. Then since this torque is "trying" to cause the object to slip, static friction comes into play and must be considered with the net torque. This is similar to our book when we try to push it. Static friction will oppose this. The difference though is that our applied torque can influence the rotation of the object, whereas our book will remain at rest. The similarity though is that static friction only comes into play when we try to make the two surfaces slide relative to each other.

In the real world though there is rolling friction. This is what usually causes things to slow down even when we aren't supplying our own torque.

The rolling without slipping condition guarantees that the constraint forces don't do work, since the point of contact with the ground is always at rest. So the cylinder must roll forever.

• What about Newton's Second Law? If friction is the only horizontal force, there must be a horizontal acceleration... – thedude Aug 9 '18 at 1:21
• This answer expresses the fully idealized situation. Real cylinders on real surfaces slow down, but that is a much harder topic. – dmckee Aug 9 '18 at 3:28
• Good aappointment – Guilherme Correa Teixeira Aug 9 '18 at 4:44

Let us assume that the initial force applied on the cylinder is at some height above the center and friction acts at the bottom. In this case of a rolling cylinder, there is no slipping due to the frictional force. So, let us imagine the cylinder as just revolving/spinning instead of rolling. The situation will be similar to a spinning a pottery wheel or a prayer wheel with an initial force/torque. The frictional force is an opposing torque continuously acting on the rotating wheel and so the speed goes down.

The frictional force acting at the bottom prevents slipping but also acts as an opposing torque thus reducing the speed of rotation.

• "The frictional force acting at the bottom prevents slipping but also acts as an opposing torque thus reducing the speed of rotation." This is only true if there are other forces acting on the object as well. – Aaron Stevens Aug 9 '18 at 2:43
• Lets say there is a box on a friction less surface. We give an impulse so that the box starts to move. It has a certain amount of kinetic energy. Now I continuously apply a force opposing the current direction of motion. The speed goes down, right? This is the same just replacing forces with torques and lateral movement with spinning. – Atchuta Srinivas Duddu Aug 9 '18 at 2:52
• You are right if there is a friction force. I am just saying that there is not always this force, even if the coefficient of friction between the two surfaces is non-zero. – Aaron Stevens Aug 9 '18 at 2:57