The plane is traveling at 288 m/s on a vertical circle of radius 622 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.
To determine the ratio of the normal force to the magnitude of the pilot's weight, we need to consider the forces acting on the pilot when the plane is traveling on a vertical circle.
When the plane is at the top of the circle, the forces acting on the pilot are the pilot's weight (mg) pointing downwards and the normal force (N) pointing upwards.
At the top of the circle, the net force is directed towards the center of the circle, providing the necessary centripetal force to keep the plane moving in a circular path. The net force is given by:
Net force = N - mg
This net force is equal to the centripetal force, which is given by:
Centripetal force = mass x (velocity)^2 / radius
Plugging in the given values, the centripetal force can be calculated as:
Centripetal force = m x (288 m/s)^2 / 622 m
To determine the ratio of the normal force to the magnitude of the pilot's weight, we can rearrange the equation for the net force:
Net force = N - mg
N - mg = m x (288 m/s)^2 / 622 m
Now, we can calculate the magnitude of the pilot's weight (mg):
mg = m x g
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Finally, we can find the ratio of the normal force to the magnitude of the pilot's weight:
Ratio = N / (mg)
Let's calculate the numerical value of the ratio using the given values and calculations above.