Animation 1 | Animation 2 | Animation 3

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Many everyday objects roll without slipping (position is given in centimeters and time is given in seconds). Restart. This motion is a mixture of a pure rotation and a pure translation. The pure rotation is shown in Animation 1, while the pure translation is shown in Animation 2. So how might we combine the two motions together so that the disk rolls without slipping?

First consider the various points on the surface of the rotating wheel. Since it has a constant angular velocity, every point has the same speed but a different velocity. Consider three special points: the top of the wheel, the wheel's hub, and the bottom of the wheel. The top of the wheel has a velocity v = ωR, and the velocity points to the right. The hub has zero velocity. And the bottom of the wheel has a velocity v = ωR to the left.

Now consider the pure translation. Every point on the wheel has a velocity v to the right.

So how do we combine the two motions together to get rolling without slipping? If the velocity of the point at the bottom of the wheel—the point that touches the ground—has a velocity of zero with respect to the ground, the wheel will not slip.

Consider the three special points again: the top of the wheel, the wheel's hub, and the bottom of the wheel. We will add the translational velocity to the rotational velocity and see what we get. The top of the wheel has a rotational velocity v = ωR to the right, which when combined with the translational velocity of v to the right gives us 2v to the right. The hub has zero rotational velocity, which when combined with the translational velocity of v to the right gives us v to the right. And finally, the bottom of the wheel has a velocity v = ωR to the left, which when combined with the translational velocity of v to the right gives us 0!

Therefore, as long as the angular velocity gives us a v that is the same v as the translation, we have rolling without slipping as in Animation 3.

Illustration authored by Mario Belloni.
Script authored by Steve Mellema, Chuck Niederriter, and Mario Belloni.