Alpengeist, in Williamsburg, Virginia, promises the world’s largest inverted (upside-down) loop, with a diameter of
106 feet (32.3 m).
(i) What is the minimum safe speed while inverted?
(ii) If the actual speed is 52.0 mph while inverted(23.25 m/s), then what is the contact force on a 60 kg inverted
person?
(i) To find the minimum safe speed while inverted, we need to consider the force of gravity acting on the rider at the top of the loop. At the top of the loop, the rider is upside down, and the gravitational force is acting downward. The minimum safe speed occurs when the normal force (the force exerted by the track on the rider) is equal to zero. At this point, the rider will not fall off the track.
To calculate the minimum safe speed, we can use the following equation:
Centripetal force = Gravitational force
For the centripetal force, we can use the formula:
Centripetal force = (mass of the rider) * (velocity squared) / (radius of the loop)
The gravitational force is equal to the weight of the rider, which is given by:
Gravitational force = (mass of the rider) * (acceleration due to gravity)
Set these two equal to each other and solve for the velocity:
(mass of the rider) * (velocity squared) / (radius of the loop) = (mass of the rider) * (acceleration due to gravity)
Cancel out the mass of the rider:
(velocity squared) / (radius of the loop) = (acceleration due to gravity)
Solve for the velocity:
velocity = sqrt((radius of the loop) * (acceleration due to gravity))
Substituting the given values:
radius of the loop = 32.3 m
acceleration due to gravity = 9.8 m/s^2
Calculate the minimum safe speed while inverted using the formula above.
(ii) To calculate the contact force on a 60 kg inverted person, we need to consider the forces acting on the rider at the top of the loop.
At the top of the loop:
- The gravitational force is acting downward.
- The normal force (the force exerted by the track on the rider) is acting upward.
- The net force is the difference between these two forces and acts toward the center of the loop.
To calculate the net force, we can use the formula:
Net force = (mass of the rider) * (velocity squared) / (radius of the loop)
Substituting the given values:
mass of the rider = 60 kg
velocity = 23.25 m/s
radius of the loop = 32.3 m
Calculate the net force using the formula above.
The contact force on the inverted person will be equal to the net force, as it needs to counterbalance the net force to keep the person on the track and prevent them from falling.
Calculate the contact force on the inverted person using the net force obtained.