A pilot flies an airplane in a vertical circular loop at a constant speed õ = 160 m/s. If the pilot’s apparent weight at the top of the loop is one-third of his true weight when he is on the ground, find the radius R of the plane’s circular path.
answer is 1960 m but i need help getting it
Apparent weight at the top of the loop is
M g - M V^2/R = (1/3) M g
Therefore
V^2/R = 2g/3
R = (3/2)V^2/g
I get twice what you say is the answer.
To find the radius of the airplane's circular path, we can start by calculating the forces acting on the pilot at the top of the loop.
At the top of the loop, the net force acting on the pilot must point toward the center of the circular path in order to keep the plane on that trajectory. So, the net force is the difference between the normal force and the weight of the pilot.
Let's denote the pilot's true weight as mg, where m is the mass of the pilot and g is the acceleration due to gravity.
We know that the pilot's apparent weight at the top of the loop is one-third of his true weight when he is on the ground. So the apparent weight is 1/3 mg.
At the top of the loop, the normal force (N) acting on the pilot is equal to his apparent weight (1/3 mg) plus his true weight (mg):
N = 1/3 mg + mg = 4/3 mg
Now, we can calculate the net force acting on the pilot at the top of the loop. In circular motion, the net force is the centripetal force:
Fc = m * v^2 / R
Where:
- Fc is the centripetal force
- m is the mass of the pilot
- v is the speed of the airplane (160 m/s)
- R is the radius of the circular path
The net force acting on the pilot is equal to the centripetal force minus the weight:
Net force = Fc - mg
Since the net force must be equal to the normal force (N) at the top of the loop, we can equate the equations:
Fc - mg = 4/3 mg
Substituting the expression for Fc:
m * v^2 / R - mg = 4/3 mg
Now, we can solve for the radius (R):
m * v^2 / R = 4/3 mg + mg
m * v^2 / R = 7/3 mg
R = (3v^2) / (7g)
Plugging in the given values:
R = (3 * (160 m/s)^2) / (7 * 9.8 m/s^2)
R = 1960 m
Therefore, the radius of the plane's circular path is 1960 meters.