Apparent Initial Acceleration due to Precession of Wheel Gyroscope?

[HakimTem]

The center of mass of a system is defined as a point that acts as if all the mass is concentrated on that one point whereby all forces may act on it. With a wheel gyroscope as an example where the center of mass is in the middle of the wheel when the wheel is spun and let go, the wheel undergoes precession whereby it rotates around its support. This comes from the nature of torque by gravity on the wheel changing its angular momentum. However, why does the center of mass gain an acceleration when it's let go to pursue a circular orbit? And which force causes such acceleration?

 

[Dale]

The force for the circular motion of the CoM comes from the support. If you spin the gyroscope on something slippery the CoM will stay in place and the support will make a circle instead.

 

[HakimTem]

So if using this as a diagram

how does the wheel initially start moving in the horizontal plane and where are the forces labeled on the diagram?

 

[Dale]

That diagram is not accurate. The chain will not hang vertically, but will make a small angle with the vertical. This small angle with the vertical will produce a restoring force like any pendulum. The restoring force will serve as the centripetal force for the circular motion of the CoM as it precesses.

 

[HakimTem]

So if the support is not supported by the ground and is allowed to move with the wheel then it would rotate around the center of mass?

 

[vanhees71]

This is indeed a heavy top. You have a torque due to gravity and the reaction force via the chain. If the wheel is spinning rapidly, you get pseudoregular precesion as described in

https://itp.uni-frankfurt.de/~hees/pf-faq/spinning-top.pdf

 

[Dale]

So if the support is not supported by the ground and is allowed to move with the wheel then it would rotate around the center of mass?

Yes. If the support cannot provide any horizontal force, but only a vertical force, then the center of mass will not move. It will just precess around the center.