A jet pilot (80 kg) flies a loop-de-loop at a constant speed of 600 m/s.
a) What minimum radius can the loop have so that the centripetal acceleration is no more than 6ag?
b) Suppose that the pilot just happened to be sitting on a spring scale while flying the jet. What will the scale read (in Newtons) when he is b) at the bottom of the loop (and flying right-side-up), and c) at the top of the loop (and flying upside-down)?
Please help ASAP!!
To find the minimum radius of the loop, we need to consider the centripetal acceleration. The centripetal acceleration is given by the equation:
ac = v^2 / r
where ac is the centripetal acceleration, v is the velocity, and r is the radius.
a) Given that the centripetal acceleration should be no more than 6 times the acceleration due to gravity (6ag), we can set up the following inequality:
ac ≤ 6ag
Substituting the equation for centripetal acceleration, we have:
v^2 / r ≤ 6ag
Substituting the given values, we have:
(600 m/s)^2 / r ≤ 6 * 9.8 m/s^2
Simplifying, we get:
360000 m^2/s^2 / r ≤ 58.8 m/s^2
Multiplying both sides by r, we have:
360000 m^2/s^2 ≤ 58.8 m/s^2 * r
Dividing both sides by 58.8 m/s^2, we get:
r ≥ 360000 m^2/s^2 / (58.8 m/s^2)
Simplifying further, we find:
r ≥ 6122.45 m
Therefore, the minimum radius for the loop is 6122.45 meters.
b) To determine the reading on the spring scale when the pilot is at the bottom of the loop (and flying right-side-up), we need to consider the forces acting on the pilot at that point. The forces include the gravitational force (mg) and the normal force (N) exerted by the scale.
At the bottom of the loop, when the pilot is right-side-up, the net force must be upward to counteract the gravitational force. This means the normal force must be greater than the gravitational force.
Therefore, the reading on the spring scale will be:
Reading = N = mg + ma
where m is the mass of the pilot and a is the centripetal acceleration.
Given that the mass of the pilot is 80 kg and the centripetal acceleration can be calculated using the formula ac = v^2 / r, where v is the velocity, we can substitute the given values to find the reading on the scale:
v = 600 m/s
r = minimum radius = 6122.45 m
Substituting these values into the formula for centripetal acceleration:
ac = (600 m/s)^2 / 6122.45 m
Calculating this, we find:
ac ≈ 5869.58 m/s^2
Substituting the values into the formula for the reading:
Reading = N = (80 kg)(9.8 m/s^2) + (80 kg)(5869.58 m/s^2)
Calculating this, we get:
Reading ≈ 469451.4 N
Therefore, the reading on the spring scale when the pilot is at the bottom of the loop (and flying right-side-up) is approximately 469451.4 Newtons.
c) To determine the reading on the spring scale when the pilot is at the top of the loop (and flying upside-down), we need to again consider the forces acting on the pilot. At the top of the loop, the net force must be downward to counteract the gravitational force. This means the normal force must be less than the gravitational force.
Using the same formula as before:
Reading = N = mg + ma
we can substitute the given values and calculate the reading:
v = 600 m/s
r = minimum radius = 6122.45 m
Substituting these values into the formula for centripetal acceleration:
ac = (600 m/s)^2 / 6122.45 m
Calculating this, we find:
ac ≈ 5869.58 m/s^2
Substituting the values into the formula for the reading:
Reading = N = (80 kg)(9.8 m/s^2) - (80 kg)(5869.58 m/s^2)
Calculating this, we get:
Reading ≈ -469171.4 N
Therefore, the reading on the spring scale when the pilot is at the top of the loop (and flying upside-down) is approximately -469171.4 Newtons. Note that the negative sign indicates that the reading is in the opposite direction of the scale's orientation, indicating that the scale is experiencing tension, rather than compression.