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: Balancing an airplane This gure is from the Federal Aviation Administration's Pilot's Handbook of Aeronautical Knowledge. It's critically important that an airplane is properly balanced before the pilot takes off. If the fuel tanks in the wings are imbalanced, a torque will roll the airplane. Likewise, too much cargo in the back of the plane will pitch the aircraft up. [ Longitudinal unbalance will cause ] either nose or tail heaviness. (a) Part of the preight planning includes calculating the weight and balance of the aircraft. The maximum total weight of this plane is 3,4001b, and the center of mass must be within an acceptable range under the wings: 7886 in from the front of the aircraft. Otherwise, the net torque on the plane will be too large. Use the following table to determine if this aircraft is safe to y. All distances are measured from the front. of the aircraft. Item W'cight [lb] Arln (in) Torque (in-lb) Aircraft Empty Weight 2100 78.3 Front Seat Occupants 340 85.0 Rear Seat Occupants 350 121.0 Fuel 450 75.0 Baggage Area 80 150.0 (b) As fuel is burned throughout the ight, the balance is continuously changing. After 3001b of fuel has been burned, and the pilot needs to apply a compensating torque with the elevators (the small wing at the rear of the aircraft) to keep the plane level. This is called setting the elevator trim. If the force is applied 27 ft from the front of the aircraft, what force is needed to create the same balance as when the fuel tank was full? 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