Problem $5.$ Reaction wheels.

Relevant learning goals:

$-$ Analyze problems involving the conservation of angular momentum.

Subskills: $1,5$

How do satellites change their orientation? This is a requirement of most satellites and spacecraft. For example, a space telescope needs to point to its targets with high precision. Firing bursts from small rocket engines is far too crude for the task.

Most spacecraft contain reaction wheels, also known as momentum wheels. They are flywheels with electric motors to spin up or slow down their rotation. When a wheel spins up in one direction, the spacecraft spins in the opposite direction.

The Hubble Space Telescope is a tube $13\mathrm{.}2$ m long with a mass of $\backslash (11,100\backslash$mathrm$\{$~kg$\}\backslash ).$ It has a reaction wheel that is a solid disk of radius of $20$ cm and a mass of $45$ kg $.$ The telescope turns at a maximum rate of $\backslash (6^\{\backslash$circ$\}\backslash )$ per minute $($very slow$).$

$($a$)$ How fast do you need to spin the reaction wheel to turn the telescope at its maximum rate? The diameter of the Hubble telescope is about $2\mathrm{.}4$ m $,$ but you can treat it as a thin rod in this problem.

$($b$)$ How many reaction wheels do you need to turn in any direction?