A stunt pilot in an air show performs a loop-the-loop in a vertical circle of radius 2.97 103 m. During this performance the pilot whose weight is 835 N, maintains a constant speed of 2.25 102 m/s.
(a) When the pilot is at the highest point of the loop determine his apparent weight.
N
(b) At what speed will the pilot experience weightlessness?
m/s
(c) When the pilot is at the lowest point of the loop determine his apparent weight.
N
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(a) To determine the pilot's apparent weight at the highest point of the loop, we need to consider the forces acting on the pilot. At this point, the pilot is at the top of the loop, moving in a circular path.
The apparent weight of an object is the force exerted by a supporting surface to counteract the force of gravity. In this case, the supporting surface is the pilot's seat.
At the highest point of the loop, the pilot's apparent weight can be calculated by considering the net force acting on the pilot. The net force is the sum of the gravitational force (mg) and the centripetal force (mv^2 / r), where m is the mass of the pilot, g is the acceleration due to gravity, v is the velocity, and r is the radius of the loop.
Since the pilot's weight is given as 835 N, we can find the mass using the formula weight = mass × gravity (835 N = mass × 9.8 m/s^2). Solving for mass, we get m = 85.2 kg.
Now we can calculate the apparent weight of the pilot at the highest point of the loop. At this point, the centripetal force must equal the gravitational force, so:
mv^2 / r = mg
Substituting the values, we get:
(85.2 kg) × (2.25 × 10^2 m/s)^2 / (2.97 × 10^3 m) = 595 N
Therefore, the pilot's apparent weight at the highest point of the loop is 595 N.
(b) To determine the speed at which the pilot will experience weightlessness, we need to consider the conditions for weightlessness. In this case, weightlessness occurs at the top of the loop when the centripetal force is equal to zero.
At the top of the loop, the net force acting on the pilot should be zero for weightlessness. Therefore:
mv^2 / r = 0
Since the radius of the loop is constant, weightlessness occurs when the velocity is zero. In other words, when the pilot momentarily comes to a stop at the highest point of the loop, he will experience weightlessness.
Therefore, the speed at which the pilot will experience weightlessness is 0 m/s.
(c) Lastly, we need to determine the pilot's apparent weight at the lowest point of the loop. At this point, the pilot is moving at a constant speed and is subjected to the forces of gravity and centripetal force.
Just like in part (a), we can calculate the apparent weight by considering the net force acting on the pilot. The centripetal force must be greater than the gravitational force at the lowest point of the loop.
Using the same mass (m = 85.2 kg), and the given radius (r = 2.97 × 10^3 m), we can calculate the apparent weight at the lowest point using the equation:
mv^2 / r = mg + apparent weight
Rearranging the equation, we get:
apparent weight = mv^2 / r - mg
Substituting the values:
(85.2 kg) × (2.25 × 10^2 m/s)^2 / (2.97 × 10^3 m) - (85.2 kg) × (9.8 m/s^2) ≈ 1375 N
Therefore, the pilot's apparent weight at the lowest point of the loop is approximately 1375 N.