The pilot of an airplane executes a loop-the-loop maneuver in a vertical circle.

Question

The pilot of an airplane executes a loop-the-loop maneuver in a vertical circle. The speed of the airplane is 330 mi/h, at the top of the loop and 450 mi/h at the bottom, and the radius of the circle is 1 440 ft. Note: His apparent weight is equal to the magnitude of the force exerted by the seat on his body.

(a) What is the pilot's apparent weight at the lowest point if his true weight is 160 lb?

(b) What is the magnitude of his apparent weight at the highest point?

(c) Describe how the pilot could experience weightlessness if both the radius and the speed can be varied.

Explanation / Answer

a)
450 mi/h = 450 * 44 / 30
= 660 ft/s

Acceleration due to the loop
= 660^2 / 1440
= 302.5 ft/s^2

Acceleration due to gravity = 32.2 ft/s^2

Total acceleration downward = 302.5 + 32.2
= 334.7 ft/s^2

True weight = 160 lb
Apparent weight = 334.7 * 160 / 32.2
= 1663.11 lb

b)
330 mi/h = 330 * 44 / 30
= 484 ft/s

Acceleration due to the loop
= 484^2 / 1440
= 162.68 ft/s^2

Total acceleration upwards
= 162.68 - 32.2
= 130.48 ft/s^2

Apparent weight
= 130.48 * 160 / 32.2
= 648.35 lb

(c) The pilot will be weightless at the top of the loop if the weight = the centripetal force, i.e., mg = mv^2/r -- in other words, whenever v^2/r = g

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