A 75 kg pilot flies a plane in a vertical loop. At the top of the loop, when the plane is completely upside-down for an instant, the pilot hangs freely in the seat and does not push against the seat belt or seat. The airspeed indicator reads 120 m/s. What is the radius of the plane's loop?
To find the radius of the plane's loop, we need to consider the forces acting on the pilot when the plane is upside down at the top of the loop. At this point, the pilot is momentarily weightless, experiencing only the gravitational force pulling them downward.
We can start by finding the gravitational force acting on the pilot at the top of the loop. The gravitational force is given by the equation:
Force_gravity = mass * acceleration_due_to_gravity
In this case, the mass of the pilot is 75 kg and the acceleration due to gravity is approximately 9.8 m/s^2.
Force_gravity = 75 kg * 9.8 m/s^2 = 735 N
Now, let's consider the net force acting on the pilot at the top of the loop. The net force is the sum of the gravitational force and the centrifugal force. The centrifugal force is the force experienced by an object moving in a circle and is given by the equation:
Centrifugal_force = mass * (velocity^2 / radius)
In this case, the mass of the pilot is 75 kg and the velocity is given by the airspeed indicator reading of 120 m/s.
Centrifugal_force = 75 kg * (120 m/s)^2 / radius
Since the pilot is momentarily weightless, the net force acting on the pilot is zero. This means that the centrifugal force and the gravitational force cancel each other out:
Force_gravity + Centrifugal_force = 0
Substituting the expressions for these forces:
735 N + 75 kg * (120 m/s)^2 / radius = 0
Now we can rearrange the equation to solve for the radius:
75 kg * (120 m/s)^2 / radius = -735 N
Simplifying the equation:
75 * 120^2 / radius = -735
Solving for the radius:
radius = 75 * 120^2 / -735
Calculating:
radius ≈ -146.56 m
Note that the negative sign indicates that the radius is directed in the opposite direction of the gravitational force. Since distance cannot be negative, we take the absolute value of the radius:
radius ≈ 146.56 m
Therefore, the radius of the plane's loop is approximately 146.56 meters.
To find the radius of the plane's loop, we can use the concept of centripetal force. At the top of the loop, the net force acting on the pilot must be directed towards the center of the loop, which provides the necessary centripetal force for circular motion.
First, let's calculate the weight of the pilot at the top of the loop. The weight of an object can be determined using the formula:
Weight = mass * gravitational acceleration
Weight = (75 kg) * (9.8 m/s^2)
Weight = 735 N
Since the pilot is hanging freely in the seat and not pushing against the seat belt or seat, the normal force acting on the pilot must be zero.
Now, let's calculate the net force on the pilot at the top of the loop. The net force can be determined using the formula:
Net force = centripetal force
At the top of the loop, the centripetal force is provided by the vertical component of the tension in the seat belt, as there is no normal force to counteract the weight. Thus, the net force can be calculated as:
Net force = Tension - Weight
Since the normal force is zero, the tension in the seat belt must be equal to the weight of the pilot.
Net force = Tension - Weight = 0
Therefore, Tension = Weight = 735 N
Now, let's calculate the centripetal force.
Centripetal force = Weight
Centripetal force = (mass * velocity^2) / radius
735 N = (75 kg * (120 m/s)^2) / radius
Now, let's solve for the radius:
radius = (75 kg * (120 m/s)^2) / 735 N
radius ≈ 156.5 m
Therefore, the radius of the plane's loop is approximately 156.5 meters.