 # Rolling Ball and Energy

A ball is on the top of a ramp so it has potential energy. Then it rolls down the ramp and has kinetic energy. Once the ball rolls to a stop, it now has less potential energy than it orginally had. Where is this energy transfered to or where does it go?

The energy goes into heating the ball and the structure it is rolling on.

If this were a frictionless system the ball would continue to roll forever. It doesn’t because friction slows it down. It would be easy to see that it is converted to heat if the ball skidded to a stop. We’re used to friction heating in those circumstances. But it’s the same phenomenon even if it’s rolling friction. The increased heat at any point on the slide would be hard to measure but it’s there.

On further reflection, some of the energy would be transferred to the earth as a whole. The earth’s momentum would increase by the same amount as the ball’s decreases.

If there was no friction the ball would never stop. It would roll forever at the kinetic energy equal to its previous potential energy.

Friction causes the ball to slow down. Frictional losses almost always take the form of heat. The air, the ground, and the ball get slightly warmer due to the rolling.

With regard to the two previous dopers:

1. If there were no friction, the ball would not be rolling. It would be sliding. Rolling motion requires friction to produce a torque, and thus, rotation.

2. If there were friction, and the ball was rolling, some of the initial gravitational potential energy would be converted to translational kinetic energy, and some would be converted to rotational kinetic energy. This energy would gradually be converted to heat as frictional forces slow the ball to a stop.

However, with rolling motion, the relative velocity between the point in contact with the surface and the surface is zero. Since there is no relative movement, there are no frictional losses between the ball and the surface. (Static friction does not produce frictional losses.)

This presents a paradox for rolling motion. Assume a ball is rolling along a perfectly smooth, horizontal surface. (Static friction must be present, but assume there are no other resistive forces, such as air resistance.) Will the ball stop?

(Answer: yes. As the ball rolls, it deforms slightly, like all objects. This deformation produces internal friction, and thus, loss of energy.)

The book went on to talk about conservation of momentum. Some must obviously be transferred to the Earth, but I’m a bit fuzzy on this part of the explanation…

Were we talking about a book?

Sorry, chopped out part of a sentence…

One paragraph up, should have read: “This presents a paradox for rolling motion, as presented by one of my general physics books.”

Looking back at this awkward sentence (again), now I remember why I cut out the phrase the first time! Neglecting the rotation of the Earth, any momentum the ball transfers to the Earth balances momentum transferred to the ball by the Earth when the ball was rolling down the ramp. So the potential energy would all end up as heat.

ok, I got some sleep…

For a ball rolling to the right and slowing down, there must be a force acting to the left. If we consider the ball and the surface to be perfectly rigid (and ignoring air resistance), there is only one horizontal force, that of static friction between the ball and the surface. It acts at the point where the ball contacts the surface.

This force acts to decelerate the ball if we look at the translational motion. However, this same force produces a torque that tends to speed the ball up if we look at the rotational motion.

Thus the paradox. Some other force must be acting on the ball.

The only way out is to realize that both the ball and the surface it is resting upon deform slightly. The rolling ball thus strikes the front of the depression it forms. This produces a counter-torque, which acts to slow the rotation of the ball.

This force also produces a slight impulse. Thus the momentum of the rolling ball may be transferred back to the Earth (as ZenBeam noted).