Problem $7\mathrm{.}20$: A gyroscope is free to move about its support point and is initially upright. It spins with an angular velocity k$^\backslash$omega $\backslash$hat$\{$k$\}$k$^,$ where $\backslash$omega is the angular speed. The axis of the gyroscope has a length ddd and negligible mass. A disk with mass MMM and radius aaa is mounted on the axis at a point d$/2$d$/2$d$/2$ from the support. The gyroscope is supported by a frictionless ball$-$bearing.

You grasp the top of the axis and rotate it until the axis is horizontal. After rotating it$,$ you hold it stationary with the axis in a horizontal orientation.

$($a$)$ How much work did you do to rotate the gyroscope axis to the horizontal position? $(5$ points$)$

$($b$)$ What is the final angular velocity of the gyroscope after the rotation? $(5$ points$)$

$($c$)$ What force must you exert on the tip of the axis to hold it stationary in the horizontal position? $(5$ points$)$

Assume that $\backslash$omega is large, so the gyroscopic precession $$p$\backslash$Omega$\_$p$$p$$ can be neglected.