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 300 mi/h at the top of the loop and 450 mi/h at the bottom, and the radius of the circle is 1,200 ft.

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

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

(c) What If? Describe how the pilot could experience weightlessness if both the radius and the speed can be varied. Note: His apparent weight is equal to the magnitude of the force exerted by the seat on his body.

Steps would be greatly appreciated!

Explanation / Answer

The speed of the plane at the top vtop = 300 mi/h = 300*0.447 = 134.1 m/s The speed of the plane at bottom is vbottom = 450 mi/h = 450*0.447 = 201.15 m/s The radius of the loop is R = 1200 ft = 1200*0.3048 = 365.76 m The true weight of the pilot w = 160 lb The mass of the pilot is m = 160*0.453 = 72.48 kg a) The apperant weight of the pilot is the normal force acting on the pilot due to seat and is given by n = mg+mv2/R at the bottom of the loop n = (72.48 kg)(9.8 m/s2) + (72.48 kg)(201.15 m/s)2/(365.76 m) n = 8728.23 N n = 1962.18 lb b) The apperant weight of the pilot is the normal force acting on the pilot due to seat and is given by n = -mg + mv2/R at the top of the loop n = -(72.48 kg)(9.8 m/s2) + (72.48 kg)(201.15 m/s)2/(365.76 m) n = 7307.62 N n = 1642.81 lb c) The pilot will experience weightlessness only if the normal force at the top becomes zero Therefore 0 = -mg + mv2/R v = Rg v = (365.76 m)(9.8 m/s2) v = 59.87 m/s v = 133.92 mi/h The pilot will experience wheightlessness if he attains a speed of 133.92 mi/h at the top of the loop of radius 1400 ft The apperant weight of the pilot is the normal force acting on the pilot due to seat and is given by n = -mg + mv2/R at the top of the loop n = -(72.48 kg)(9.8 m/s2) + (72.48 kg)(201.15 m/s)2/(365.76 m) n = 7307.62 N n = 1642.81 lb c) The pilot will experience weightlessness only if the normal force at the top becomes zero Therefore 0 = -mg + mv2/R v = Rg v = (365.76 m)(9.8 m/s2) v = 59.87 m/s v = 133.92 mi/h The pilot will experience wheightlessness if he attains a speed of 133.92 mi/h at the top of the loop of radius 1400 ft

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