The half precesses slower than the whole. I'm not sure if this is simply because it has a lower center of gravity, or because the ratio of inertia of the heavier whole globe relative to the torque applied by the string is greater compared to the lighter half version. It be swell if the physics experts might weigh in on this please?
Let’s consider the case of a simple top spinning on a surface with the tip position fixed so we can ignore all the complications added by a player’s motion of the string etc. A freshmen physics text will provide an analysis of a spinning top in a gravitational field resulting in an equation for the precession rate of the top in terms of the following parameters:
m= total mass of the top
I= moment of inertia about the axis of rotation of the top (basically a representation of the distribution of the mass about the axis of rotation)
r= distance of the center of mass (which lies on the axis of rotation) from the fixed point of rotation
w= angular rate of rotation (angular speed) about the top’s axis (spinning speed)
P= angular rate of rotation (angular speed) about the direction of gravity or the rate of Precession
g= the acceleration due to gravity (just a constant due to the earth in our case)
The equation is P = (m * r * g) / (I * w)
where the asterisk “ * ” means multiply and the slash “ / “ means divide.
If the mass or the distance to the center of mass is increased, the precession rate gets faster.
If the mass or the distance to the center of mass is decreased, the precession rate gets slower.
If the moment of inertia or the spinning speed increases, the precession rate gets slower.
If the moment of inertia or the spinning speed decreases, the precession rate gets faster.
For the case of your half and full ball assemblies, several of the parameters change making the problem more complicated.
1)Adding the second half increases the total mass m which increases the precession rate for the full ball.
2)Adding the second half increases the height of the center of mass so r increases which increases the precession rate for the full ball
3)Adding the second half increases the moment of inertia which decreases the precession rate for the full ball
4)The added moment of inertia due to the second half reduces the spinning speed w for the same number and strength of regeneration pulls so most likely w is lower for the full ball which means the precession rate for the full ball is increased.
Thus the change of adding the second half adjusts the relevant parameters in such a way that some tend to increase the precession rate (1,2,4) and others tend to decrease the precession rate (3). In your particular observations, for the full ball assembly vs the half ball, the rate increasing parameters win out making the full ball precess more quickly. With some regenerating effort you may be able to get the full ball spinning fast enough such that it precesses slower than the half ball.
