Rolling Ball III A

Classical Mechanics Level 2

A solid ball with a moment of inertia of $I=\frac{2}{5}mr^2$ is initially moving with a velocity of $v_i$ and it is sliding with no rotational motion at $t=0$ on flat ground (which you can assume to be extended to infinity).

We know the kinetic friction does work on the system. As we are familiar with frictional forces, they are not energy conservative.

In the case of the rolling ball, what is the role of kinetic friction in terms of Translational KE and Rotational KE during the slow down phase?

note: (assume the ball to be rigid and the ground too, so you don't have to consider rolling friction. KE is Kinetic Energy.)

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It does zero net work, and it just keeps converting TKE to RKE. It will keep doing negative work until the ball slows to a stop. It increases RKE and TKE. It does negative work on TKE but positive work on RKE, until the ball stops slipping.

1 solution

Eduardo Ichige
Jun 30, 2019

Let's divide this problem in three moments:

(1) when the ball has only TKE;

(2) when the ball has some TKE and some RKE, but is still sliding;

(3) when the ball das some TKE and some RKE and is not sliding anymore. At this point, there is no more frictional force, whis will be explained bellow.

From moment (1) to moment (2) the frictional force is doing work on the ball, converting TKE into RKE. This means that TKE is going down (negative work) and RKE is going up (positive work).

When we reach moment (3) there is no more sliding, this means that there is no relative velocity between the ball and the ground at the point of contact (the ball as a whole is still moving relative to the ground, but its part that is touching the ground is not). This point of contact is where the kinectic frictional force is applied. Since kinectic frictional force is dependent upon the velocity of where it is applied, we can conclude that kinectic frictional force is zero.

Because there is no more kinectic fricitonal force after moment (3) the ball will continue to roll and not come to a stop. This doesn't mean that the inicial energy is conserved, just imagine that the amount of negative work on TKE in greater than the amount of positive work on RKE

Rolling Ball III A

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