# Torque and angular acceleration with bicycle wheel

This might be a simple problem for many of you. However, please help me understand it too. I have been looking trough a lot of materials online, and I still have the following questions, that would help me a lot.

I saw this video on youtube: see wheel in action

If we consider the following model:

Please clarify for me the following questions:

1. I believe the gravity is not pulling the wheel down because the torque resulting from multiplying gravity by the l vector is pointing sideways. In other word, the down-pointing gravity vector is transformed into a vector pointing sideways and spinning the wheel around the string. (Is this how force multiplication works? do the multiplied factors transform into the resulting force?)

2. If the wheel would keep a constant angular velocity(assuming no friction) the wheel would spin indefinitely without rising or falling. What would happen if the angular velocity would be increased while spinning? would the wheel rise (ie the free floating end would rise)?

3. Mathematically what causes the wheel to fall when the angular velocity decreases? The only changing quantity is L, which depends on the speed of the spinning wheel. But the direction of L is not changing. And from (1) above, the gravity should not be pointing downwards. So why does the wheel fall when L gets smaller(by wheel falling I mean the free floating end goes down, until it is under the string)

1) 'Gravity is not pulling x down' is a rather confusing way to think about it (as it's always there), but you are right. What's happening is the cross-product, which requires two vectors as an argument. The result is a vector that is perpendicular to both initial vectors. Of course being perpendicular to both still leaves two directions (check it yourself!), but the cross-product has been defined in such a way that it specifies only one direction. When the professor changes the direction of the spin, he changes $\vec L$ to point to the other direction; as a result the result of the cross-product switches direction and the wheel turns the other way.