A jet pilot takes his aircraft in a vertical loop. If the jet is moving at a speed of 1100km/h at the lowest point of the loop:
A)Determine the minimum radius of the circle so that the centripetal acceleration at the lowest point does not exceed 5.5g's
B)Calculate the 68kg pilot's effective weight (the force with which the seat pushes up on him) at the bottom of the circle.
C)Calculate the 68kg pilot's effective weight (the force with which the seat pushes up on him) at the top of the circle (assume the same speed)
v =1100 km/h = 305.6 m/s.
a =v^2/R => R = v^2/a = (305.6)^2/5.5•9.8 = 1732 m
R > 1732 m.
The magnitude of the effective weight = the magnitude of the normal force N
bottom: N = m•g – m•a = m• (g - 5.5•g) = - 4.5•m•g
W = - N = 4.5•m•g =4.5•68•9.8 =2998.8 N.
top: N = - m•g – m•a = -m• (g+a) = -m• (g+5.5•g) ,
W = - N = 6.5•m•g =6.5•68•9.8 =4331.6 N.
To solve these problems, we need to apply the concepts of centripetal acceleration and effective weight.
A) To determine the minimum radius of the loop so that the centripetal acceleration at the lowest point does not exceed 5.5g's, we can use the equation:
Centripetal acceleration = (velocity^2) / radius
We are given that the velocity of the jet is 1100 km/h at the lowest point. To convert this to m/s, divide by 3.6 (1 km/h = 1000 m/3600 s). Thus, the velocity of the jet at the lowest point is (1100 km/h) / (3.6) = 305.56 m/s.
We also know that the centripetal acceleration should not exceed 5.5g's. One g is equal to the acceleration due to gravity, which is approximately 9.8 m/s^2. Thus, 5.5g's would be 5.5 times the acceleration due to gravity.
Let's denote the minimum radius of the circle as "r". We can now plug in the values into the centripetal acceleration equation:
5.5g's * (9.8 m/s^2) = (305.56 m/s)^2 / r
Solving for r, we get:
r = (305.56 m/s)^2 / (5.5 * 9.8 m/s^2)
B) To calculate the pilot's effective weight at the bottom of the circle, we need to consider the net force acting on the pilot. The net force is the sum of the gravitational force and the normal force.
At the bottom of the loop, the normal force provides the centripetal force. Thus,
gravitational force + normal force = centripetal force
The gravitational force can be calculated using the equation:
Gravitational force = mass * acceleration due to gravity
The centripetal force can be calculated using the equation:
Centripetal force = mass * centripetal acceleration
In this case, the centripetal acceleration is equal to the acceleration due to gravity (g), as the pilot is not experiencing any additional acceleration due to the circular motion.
Let's denote the effective weight (normal force) as "N". We can now set up the equation:
mass * acceleration due to gravity + N = mass * acceleration due to gravity
Simplifying, we find:
N = mass * acceleration due to gravity
C) At the top of the circle, the normal force and gravitational force act in opposite directions. So the effective weight at the top can be found as the difference between the gravitational force and the normal force. Using a similar process as in part B, we can find the effective weight at the top of the circle.