A pilot, whose mass is 70.0 kg, makes a loop-the-loop in a fast jet. Assume that the jet maintains a constant speed of 325 m/s and that the radius of the loop-the-loop is 5.922 km. What is the apparent weight that the pilot feels (i.e., the force with which the pilot presses against the seat) at the bottom of the loop-the-loop?
What is the pilot's apparent weight at the top of the loop-the-loop?
To find the apparent weight of the pilot at the bottom and top of the loop-the-loop, we need to consider the forces acting on the pilot.
At the bottom of the loop, the pilot experiences two forces: their weight (mg) and the normal force (N) exerted by the seat. The net force acting on the pilot is the sum of these two forces.
At the top of the loop, the pilot also experiences the weight (mg) and the normal force (N) exerted by the seat. However, in this case, the net force acting on the pilot is the difference between these two forces.
To calculate the apparent weight at the bottom of the loop, we can use the following equation:
Net force at bottom = Weight - Normal force at bottom
The normal force at the bottom of the loop is equal to the sum of the weight and the centripetal force:
Normal force at bottom = Weight + Centripetal force at bottom
To calculate the centripetal force at the bottom of the loop, we can use the formula:
Centripetal force at bottom = mass * velocity^2 / radius
Substituting the given values into the formulas, we can calculate the apparent weight at the bottom of the loop:
Apparent weight at bottom = Weight - (Weight + Centripetal force at bottom)
Now, let's calculate the values.
Weight = mass * gravitational acceleration
= 70.0 kg * 9.8 m/s^2
= 686 N
Centripetal force at bottom = mass * velocity^2 / radius
= 70.0 kg * (325 m/s)^2 / 5922 m
= 1290 N
Normal force at bottom = Weight + Centripetal force at bottom
= 686 N + 1290 N
= 1976 N
Apparent weight at bottom = Weight - Normal force at bottom
= 686 N - 1976 N
= -1290 N
Therefore, the pilot's apparent weight at the bottom of the loop-the-loop is -1290 N. The negative sign indicates that the force is directed upward, away from the seat.
To find the apparent weight at the top of the loop, we can use the same approach:
Net force at top = Weight - Normal force at top
The normal force at the top of the loop is equal to the difference between the weight and the centripetal force:
Normal force at top = Weight - Centripetal force at top
To calculate the centripetal force at the top of the loop, we can use the same formula as before, but with the opposite sign:
Centripetal force at top = -mass * velocity^2 / radius
Now, let's calculate the values.
Centripetal force at top = -mass * velocity^2 / radius
= -70.0 kg * (325 m/s)^2 / 5922 m
= -1290 N
Normal force at top = Weight - Centripetal force at top
= 686 N - (-1290 N)
= 1976 N
Apparent weight at top = Weight - Normal force at top
= 686 N - 1976 N
= -1290 N
Therefore, the pilot's apparent weight at the top of the loop-the-loop is -1290 N, which is the same as at the bottom. Again, the negative sign indicates that the force is directed upward, away from the seat.