At the top of a looped section of roller coaster track, the car and rider are completely upside down. Engineers calculated that the car may travel a
minimum speed of 14.3 m/s to keep riders from faling out of the car at the top of the loop. What is the radius of the loop?
20.9 m
17.4 m
18.9 m
22.5 m
To determine the radius of the loop, we can use the centripetal force and the gravitational force acting on the rider at the top of the loop.
At the top of the loop, the net force acting on the rider should be equal to the centripetal force to provide enough inward acceleration to keep the rider from falling out of the car. The net force is given by the difference between the gravitational force and the normal force.
The gravitational force pulling the rider downward is given by mg, where m is the mass of the rider and g is the acceleration due to gravity (approximately 9.8 m/s^2).
At the top of the loop, the normal force acting on the rider is zero since it is completely upside down. Therefore, the net force is equal to the gravitational force alone.
The centripetal force is given by (mass of the rider) × (minimum speed)^2 divided by the radius of the loop.
Setting the gravitational force equal to the centripetal force, we can solve for the radius:
mg = (mass of the rider) × (minimum speed)^2 / radius
Simplifying the equation:
radius = (mass of the rider) × (minimum speed)^2 / (g)
Now we can calculate the radius using the given minimum speed of 14.3 m/s and the acceleration due to gravity of 9.8 m/s^2, and by assuming the mass of the rider is 1 kg:
radius = (1 kg) × (14.3 m/s)^2 / (9.8 m/s^2)
Calculating this equation yields approximately 20.9 m.
Therefore, the radius of the loop is approximately 20.9 m.