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.

The pilot is sitting in the plane, therefore there is a normal force N exerted by the seat and his weight. There is also a centripetal force with value

[tex]\frac{mv^2}{r} N[/tex], where m is the mass of the pilot, v is the velocity, and r the raidus of the circle. We need to solve N

Now 2nd law is takes the following form:

[tex]N - mg =\frac{mv^2}{r}=>N =\frac{mv^2}{r}+mg[/tex]

we have m = 72.5748 [Kg], r = 60.96 [m], [tex]v_{top} = 134.112, v_{bottom}= 201.168[/tex]

we can now replace the velocities for items a) and b)

a) [tex]N = (72.5748*(201.168)^2)/(60.96)+72.5748*9.8 = 48.890 [N]\\[/tex]

b) [tex]N = (72.5748*(134.112)^2)/(60.96)+72.5748*9.8 = 22.124 [N]\\[/tex]

c) If N is 0 we acheive weightlessness [tex]\frac{mv^2}{r} = mg=>\frac{v^2}{r} = g[/tex], let r be the same (60.96) then v should be [tex]v=\sqrt{r*g}=\sqrt{60.96*9.8}=24.44 [m/s][/tex]

obviously.. any plane would stall at that velocity, but weightlessness is achieved