Hi there, can you walk me through each part of the problem below? Can you help me understand the necessary steps to solve each part? Please also provide a response to each prompt so I can compare with my own solutions.

I have added a few formula reference that may be needed for the problem below. (Any unknown information can be found by searching on the internet/making reasonable or general assumptions)

Practice Problem: Reaction Wheels 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 ywheels 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 132m long with a mass of 11,100 kg. It has a reaction wheel that has a radius of 20 cm and a mass of 45 kg. The telescope turns at a maximum rate of 6 per minute (very slow). (a) Where do you ideally place the reaction wheel for maximum effect? Why? (b) How fast do you need to spin the reaction wheel to turn the telescope at its maximum rate? (c) (Optional) How many reaction wheels do you need to turn in any direction? Work and Energy: Kinetic energy: K = =my Work: W = F . dr Work - Energy theorem: Wnet = AK Potential Energy (conservative Fc): W. = -AUc Conservation of Energy: Wnc = AK + AU Gravitational potential energy near the surface of the earth: U. = mgy Gmim2 Gravitational potential energy: UG = - r Spring potential energy: Usp = ka2 (for proper choice of coordinate system) Momentum and Impulse: Momentum: p = mu, Impulse: J = F dt ti Impulse-Momentum Theorem: J = Ap Conservation of Momentum: Ptot, f = Ptot, i (with no external forces)General: r(t) = x(t) ity(t) j+ z(t)k, dr du dar v (t ) = at ' a(t) = dt dt Newton's 2nd Law: For a an object of mass m: _ Fext = Fnet = ma Common Forces: Force due to gravity near surface of the earth: F , = mg Universal gravitation: |Fal = Gmim2 72 Spring force: Fsp = kell- lol in the direction to restore equilibrium Static friction: Kinetic friction: |FS | = HK N Kinematics with constant acceleration: u(t) = vo ta . (t -to) r(t) = ro+ vo . (t -to)+ za. (t-to)2 For a single dimension / axis x, we can also write: Uf x = Vox + 2ax(x5 -20) Simple Harmonic Oscillators: For a spring-mass system (k, and m): x(t) = Acos(wt + 4) 2 7T Where: W = T and T is the period of motion Circular Motion: de Arc Length: s = re Angular velocity: W = dt v = wr Centripetal acceleration: ac = = warReference for mements of inertia Axis Hoop about cylinder axis l = MR2 Axis Solid cylinder (or disk) about " cylinder axis , = ?- 2 Axis _ Thin rod about axis through i center J. to length it? \"'th Axis Solid sphere 2F? about any diameter _ zone I 5 Axis Hoop about t. any diameter Annular cylinder {or ring} about cylinder axis :=%tsi+s) Solid cylinder (or disk} about " oentral diameter Thin rod about axis through one i end i to length _ MP ' T Axis Thin . 2R spherical shell about any diameter l = ans? 3 Axis Slab about i axis through center b a If : M 12 Figure 10.21] Values of rotational inertia for common shapes of objects