Homework problem dealing with centripetal force?

Question

Pilots of fighter planes are subjected to large centripetal accelerations during high-speed turns. Because of these accelerations, the pilots are subjected to forces are greater than their body weight, leading to an accumulation of blood in the abdomen and legs. As a result, the brain is starved for blood, and the pilot can ("black out"). To appreciate the forces that a fighter pilot must endure, consider the magnitude of the normal force that the pilot's seat exerts on him at the bottom of a dive. The plane is traveling at 236 m/s on a vertical circle of radius 738 m. Determine the ratio of the normal force to the magnitude of the pilot's weight. For comparison, note that black-out can occur for ratios as small as 2 if the pilot is not wearing an anti-G suit.

My work:

Ratio = FN/W
FN=ma ---> m(v^2/r)
W=mg

so ((mv^2)/r)/(mg) the m's cancel so (236^2)/738/9.8= 7.7009

but that answer wasn't correct. I need help and any help would be appreciated. Thanks.

Answer 1

Alex had the right idea, but I wanted to make a small correction. At the bottom of the loop, there are two forces acting on the pilot. One is the upward force supplied by the seat, and the second force is the pilot's weight, acting downward. The sum of these two forces must equal the centripetal force which is making the pilot go in a circle. So we have the following:

FN + (-mg) = m(v^2/r)
FN = m(v^2/r)+mg
FN/W=(m(v^2/r)+mg)/mg

You end up ADDING the weight to the centripetal force, rather than subtracting. An easy way to visualize this is to realize that more gravity will INCREASE the upward force exerted on the pilot by the seat at the bottom of the loop, rather than decreasing it....

Answer 2

The plane is traveling on a vertical circle. So gravity force modifies the normal force of the seat. You`re asked for the ratio when the plane is at the bottom of the circle, so the normal force is actually the centripetal force minus the weight.

FN/W=(m(v^2/r)-mg)/mg

good luck.

Answer 3

The circle is vertical, so the TOTAL force involves both centripetal force and gravity.

Answer 4

Fc = Fg = G*Ms*Me/R² = 6.67E-11*1.989E30*5.98E24/(1.5E11)² = 3.526E22 N Period not required, but can be calculated from T² = 4π²R³/(GMs)