Problem $4.$ Balancing an airplane.

Relevant learning goals:

Calculate the torque due to a force about a reference point or axis.

Subskills: $2,3$

This figure is from the Federal Aviation Administration's Pilot's Handbook of Aeronautical Knowl$-$

edge. 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$.$

$($a$)$ Part of the preflight planning includes calculating the weight and balance of the aircraft. The

maximum total weight of this plane is $3,400$lb$,$ and the center of mass must be within an

acceptable range under the wings: $78-86$ 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 fly. All distances are measured from the front of the aircraft. Note that the book uses

imperial units $($pounds of force and inches$).$

Item

Aircraft Empty Weight $(2100$ lbs$,$ Arm $=78\mathrm{.}3$in$)$

Front Seat Occupants $(340$ lbs$,$ Arm $=85$ in$)$

Rear Seat Occupants $(350$ lbs$,$ Arm $=121$ in$)$

Fuel $(450$ lbs$,$ Arm $=75$in$)$

Baggage Area $(80$ lbs$,150$ in$)$

Total

$($b$)$ As fuel is burned throughout the flight, the balance is continuously changing. After $300$ lb 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?