Torque and angular acceleration with bicycle wheel

Question

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)

Answer 1

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.

It's important to note that gravity isn't doing any work or is being worked on - after all, the wheel doesn't go up or down.

2) If spin velocity was higher, it would turn quicker; the axis of the wheel would stay horizontal.

3) After a while, friction kicks in and slows down the wheel to the point where the torque can no longer support the wheel, just like how a top starts waggling and falls down when it slows down too much.

If you find this hard to follow let me know; I'll try to simplify.

Answer 2

First the diagram is misleading, it shows torque produced at point where wheel axis is attached to string while it is at centre of mass of wheel-axis which is close to wheel. For simplicity, we can assume centre of mass at centre of wheel. Now comes to questions asked.

1. This is important question because it makes look like that something magical is happening which defying gravity, otherwise weight of wheel pulls down and its axis in the line of string. Reason is that weight applied at wheel which produces torque, that changes angular momentum perpendicular to both gravity and angular momentum of wheel due to spinning. Why, because torque is cross product of radial distance which is direction of angular momentum and weight of wheel which is downward or vertical direction.

This change in angular momentum changes direction of spinning of wheel. At this point, again wheel produces torque which again changes direction of angular momentum of wheel. But this force of gravity now countered by moment of force which is force times radial distance at the string end on axis in same direction as gravity, so it raises wheel end. If wheel is not spinning, then wheel simply comes down, but due to spin, produced torque first changes its direction of spin that pushes wheel to move in direction of torque. This motion of wheel perpendicular to its spinning is caused by force in direction of torque. This force coupled with axis arm produces torque in downward direction at string end.

2. If the angular speed of wheel's spinning is increase then axis if wheel lies more flat or horizontally, otherwise it inclined to upward or vertical direction. Because increase in angular momentum cause increase in torque, and only way to do that is increase arm length of torque.

3. As the speed of wheel's spin goes down, its free end rises upward. So circle of precessional motion becomes smaller that makes angular speed of precessional motion constant. And as circle becomes smaller, the torque also becomes smaller as speed of spinning reduces.