Monthly Mechanics: How Fast?

Last month I asked a simple question with a not-so-simple answer. If your car is slinking along the road at, say, 5 feet per second (about 3.4 mph), how fast does the car’s tire move?

Solution: In order to make a complete revolution, the outside of a wheel needs to make a complete circle, so it moves the circumference of that circle. To make the math easier, let’s say your car tire has a circumference of 1 foot. (Maybe your car is a shopping cart, or a Radio Flyer wagon.) That means if the car is traveling 5 feet per second, the tire is making 5 revolutions per second.

Wheel1

It helps if you ignore the road and the car for a moment, and think about the wheel all by itself. Every point on the outside of the wheel moves 5 feet per second. The top of the wheel is moving forward; the bottom is moving backwards; the left side is moving up; the right side is moving down. But they’re all moving at a speed that allows the wheel to make 5 revolutions per second, or one revolution every 0.20s.

Wheel2

If you think about this a little more, you’ll realize it doesn’t matter how big or small the wheel is! A wheel with a circumference of 10 feet (monster truck tires) would make half a revolution per second, but that still comes out to 5 ft/s.

Now let’s bring the car and the road back into the picture. The whole car is moving 5 ft/s along the road, so the axle is moving forward at 5 ft/s as well. At the same time, the top of the wheel is moving ahead of the axle, and the bottom of the wheel is moving behind the axle. The top goes 5 ft/s faster than the axle, and the bottom goes 5 ft/s slower.

Wheel3

Putting two and two together (or five and five, in this case), the top of the wheel is moving forward at 10 ft/s, twice as fast as the car is going. And the bottom of the wheel is moving… zero ft/s. That’s right, the part of a tire that touches the road isn’t moving at all.

But just for an instant. A moment later, a different part of the wheel is touching the floor, and the part that was touching before is now in the air. This is precisely what makes rolling motion so efficient. The friction between the wheel and the road is limited to that tiny portion of the tire that’s instantaneously in contact. It’s not sliding against the road and it doesn’t have to; it just needs to tip over to the next bit of tire, and that moves the whole car forward.

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