# What forces are responsible for the circular motion of the center of mass of a gyroscope-like setup in precession?

Suppose we have a disk (of mass $$M$$) connected to a rod (of mass $$m$$) attached to a fixed pivot. (Note that initially I thought of the rod hanging off a rope, but I realized it would be better to assume the rod is attached to a fixed pivot). The rod is horizontal along the $$x$$-axis, with the disk in the $$+x$$-direction, and the disk is spinning with angular momentum $$\vec{L} = L\,\vec{e}_{x}$$. Gravity imposes a force on the rod and disk downward in the $$-z$$-direction, but due to the way the torque is imposed, the net result is that the setup precesses about the $$z$$-axis.

We have three forces at play: the gravitation force on the disk, the gravitation force on the rod, and the pivot force keeping one end of the rod in place (which I label as $$\vec{T}$$).

Now the center of mass of the rod + disk is somewhere at $$x > 0$$ in the image depicted. Since the center of mass does not fall, we must have $$T = Mg + mg$$ in terms of magnitudes. Now I have two questions:

1. Doesn't the center of mass of the rod + disk move in circular motion around the $$z$$-axis? If so, doesn't there have to be an additional force from somewhere causing this motion? What is this force and what am I failing to account for here?

2. Since the center of mass itself moves circularly around the $$z$$-axis (due to precession), doesn't that meant there is an additional angular momentum component pointing upward? Where does this $$z$$-angular momentum come from? Even if it is constant, there had to have been an initial torque coming from somewhere, so where would that upward-pointing torque come from? For concreteness, imagine a demo where you hold the spinning bicycle wheel in place, and then let go. When you let go, the fact that the center of mass starts to move around the $$z$$-axis means there had to have been an initial vertical torque. What force is responsible for that initial torque and how can we understand it?