A stunt pilot in an air show performs a loop-the-loop in a vertical circle of radius 3.70 103 m. During this performance the pilot whose weight is 740 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
(a) When the pilot is at the highest point of the loop, his apparent weight can be determined using the equation:
Apparent Weight = Weight + Centripetal Force
The centripetal force is provided by the pilot's normal force, which is equal to the weight when there is no net force in the vertical direction.
Apparent Weight = Weight + Weight
Apparent Weight = 2 * Weight
Therefore, the apparent weight at the highest point of the loop is 2 times the pilot's weight.
Apparent Weight = 2 * 740 N
Apparent Weight = 1480 N
So, the pilot's apparent weight at the highest point of the loop is 1480 N.
(b) The pilot will experience weightlessness when the net force on him is zero. At the highest point of the loop, the net force acting on the pilot is equal to the centripetal force.
Centripetal Force = Weight
Using the equation for centripetal force:
(mass x speed^2) / radius = Weight
mass x speed^2 = Weight x radius
mass x (speed^2/radius) = Weight
mass = Weight / (speed^2/radius)
mass = 740 N / ((2.25 x 10^2 m/s)^2 / (3.70 x 10^3 m))
Plugging in the values and calculating:
mass = 740 N / (506.25 / 3700)
mass = 740 N / 0.1365
mass = 5414.85 kg
So, the mass of the pilot is approximately 5414.85 kg.
(c) At the lowest point of the loop, the pilot's apparent weight can be determined using the same equation as in part (a):
Apparent Weight = Weight + Centripetal Force
However, at the lowest point of the loop, the centripetal force is provided by the pilot's normal force, which is equal to the pilot's weight plus the centripetal force.
Centripetal Force = Weight + Weight
Centripetal Force = 2 * Weight
Using the equation for centripetal force:
(mass x speed^2) / radius = 2 * Weight
mass x speed^2 = 2 * Weight x radius
mass = (2 * Weight x radius) / speed^2
mass = (2 x 740 N x (3.70 x 10^3 m)) / (2.25 x 10^2 m/s)^2
Plugging in the values and calculating:
mass = (2 x 740 N x 3.70 x 10^3 m) / (5.0625 x 10^4 m^2/s^2)
mass ≈ 0.9775 kg
So, the mass of the pilot is approximately 0.9775 kg.
Therefore, the apparent weight of the pilot at the lowest point of the loop is:
Apparent Weight = mass x gravity
Apparent Weight = 0.9775 kg x 9.81 m/s^2
Apparent Weight ≈ 9.58 N
So, the pilot's apparent weight at the lowest point of the loop is approximately 9.58 N.
To solve these questions, we can use the concept of centripetal force and apparent weight.
(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 the highest point, the pilot will experience two forces - the gravitational force and the centripetal force. The centripetal force is responsible for keeping the pilot moving in a circular path.
The centripetal force is provided by the net force acting on the pilot, which is equal to the difference between the gravitational force and the apparent weight.
At the highest point, the apparent weight will be the minimum, as it counteracts the gravitational force. Therefore, the net force is equal to zero.
The net force can be calculated using the equation:
Net force = gravitational force - apparent weight
At the highest point, the net force is zero. So we can write:
0 = gravitational force - apparent weight
Rearranging this equation, we get:
apparent weight = gravitational force
Therefore, the apparent weight at the highest point is equal to the pilot's weight, which is 740 N.
(b) To determine the speed at which the pilot experiences weightlessness, we need to consider the forces acting on the pilot. At the point of weightlessness, the pilot is not experiencing any apparent weight or feeling any force acting on him/her.
At this point, the net force acting on the pilot is equal to zero. The centripetal force is still acting, but there is no apparent weight.
Using the same equation as above, we can write:
0 = gravitational force - apparent weight
Since the gravitational force is acting downward, the apparent weight is equal to zero when the pilot experiences weightlessness.
To find the speed at which the pilot experiences weightlessness, we can set the apparent weight to zero and solve for the centripetal force.
0 = gravitational force - 0
Therefore, the speed required for the pilot to experience weightlessness is when the centripetal force provided by the circular motion exactly cancels out the gravitational force.
(c) At the lowest point of the loop, the apparent weight will be at its highest value. This is when the pilot is being pushed into the seat with the gravitational force.
To determine the apparent weight at the lowest point, we use the same equation as above:
Net force = gravitational force - apparent weight
At the lowest point, the net force is the centripetal force, which is the sum of the gravitational force and the apparent weight.
Centripetal force = gravitational force + apparent weight
Therefore, the apparent weight at the lowest point is equal to the sum of the gravitational force and the centripetal force.
Apparent weight = gravitational force + centripetal force
We can calculate the centripetal force using the formula:
Centripetal force = mass * (velocity^2) / radius
Substituting the given values, we get:
Centripetal force = (740 N) + (740 N)
Therefore, the apparent weight at the lowest point is 1480 N.
you bad
a. weight=mg-mv^2/r
b. set weight to zero, and solve for v.
c. weight=mg+mv^2/r