Asked by: Steve Collier

To understand angular momentum, you must understand torque. Torque is what you do to a door when you push it open, or turn the knob. Anytime there is something that swings or spins about a fixed point, a torque must be applied to change the motion of that object. Torques are applied to a lever arm which is the perpendicular distance from the axis of rotation to the line that the force is acting on, therefore the lever arm is a vector like the force. Torque is then defined as the cross product of the lever arm and the force. An alternate definition of torque is the rate of change of angular momentum. Because torque is a cross product of vectors, the direction of the torque is perpendicular to the directions of the force and lever arm. The direction for torque and therefore angular momentum is given by the right-hand-rule.

The reason that you stay up on a bike is that angular momentum, like regular momentum, must be conserved if no external torques act on the object. Because angular momentum is a vector, not only must its magnitude be conserved, but also its direction. This implies that any change in the orientation of the object will change the vector for angular momentum. Therefore, tops stay up because the angular momentum wants to stay conserved in the same direction.

When you are riding a bike forward, the right hand rule gives the direction of angular momentum to be to the left, perpendicular to the wheel. This direction does not want to change, therefore the wheel wants to stay upright and it makes the bike very ridable. Have you noticed it is harder to ride a slow bike? Because the wheels are moving slow, the angular momentum is less and the direction of rotation is easily changed. I have seen a bicycle built with an extra wheel that did not touch the ground so that it could be spun forwards or backwards so that it either added to or subtracted from the total angular momentum of the bike. When the extra wheel was spun in the same direction as the other wheels, the bike was extremely easy to ride even at low speeds. However, when the extra wheel was spun backwards, the bike became almost impossible to ride because the vectors for angular momentum cancelled each other out. It was like trying to balance a bike that was not moving.

I hope this answers the question about riding a bike. Could we possibly make intermediate training wheels that do not touch the ground, but add to the ridability of a bike?

Answered by: Matthew Allen, B.S., Physics/Calculus Teacher St. Scholastica Academy

'The mathematician's patterns, like the painter's or the poets, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.'**Godfrey Hardy**

(*1877-1947*)

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