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# Rolling

Wheels are a marvelous invention that allow us to zip around a lot faster than we would normally with our own two legs. Whatever cave-creature that invented the wheel should certainly be given the Nobel prize. On the other hand, they did choose a pretty stupid spelling for their invention.

A lot of mechanics problems involve wheels. There interesting because they couple translational motion with rotational motion. If a wheel is rolling with a velocity v, what is it's angular velocity? We can compute that by considereing the figure below.

The red dot marks a fixed spot on the rim of the wheel. In the first snapshot, the red dot is next to the ground. In the second, it has rotated 180 degrees and is now at the top. Finally at time T, it's back at the ground again. The wheel has done one revolution and in the same time, the center of the wheel has gone the wheel's circumference , where r is the radius of the wheel. Now the angular velocity . That is, it has done radians in the time it takes to make one revolution, T. So the velocity

Imagaine that! It's our old friend . But this means something quite different here. The v here is the center of mass velocity of the wheel. Our old friend refers to a different v, the velocity of the rim of the wheel when the center isn't moving. OK, so what do we make out of all of this? We can think of rolling as combining a pure translation of the wheel at center of mass velocity with rotational motion with angular velocity . This then tells us what the speed of the wheel is at various places on it's rim. For example, at the top of the wheel, we add to the velocity due to rotational motion . This gives . At the bottom of the wheel, right next to the ground, these two effects cancell, and so the velocity of the wheel at the ground is zero! This makes sense, because if it where different from zero, the wheel would be sliding along the ground, which we are not allowing. The following is a sketch of the velocity vectors for several points along the rim of the wheel.

Notice that it is identical to the velocity if the wheel was pivoted at the bottom. So instantaneeously a rolling wheel looks as if it's pivoted.

Next: Kinetic energy Up: Angular Momemtum and Torque Previous: Conservation of angular momentum

Joshua Deutsch
Sun Feb 23 15:54:50 PST 1997