A carnival clown, 80 kg, rides a motorcycle down a ramp and then up and around a large, vertical loop. If the loop has a radius of 18 m, what is the force that the track exerts on the rider at the TOP of the loop if the clown is traveling at an impressive speed of 15 m/s?
To calculate the force exerted by the track on the rider at the top of the loop, we need to consider the forces acting on the rider. At the top of the loop, there are two forces involved: the gravitational force pulling downward and the normal force exerted by the track pushing upward.
Step 1: Calculate the gravitational force:
The gravitational force acting on the rider can be calculated using the formula:
F_gravity = m * g
where:
m = mass of the rider (80 kg)
g = acceleration due to gravity (approximately 9.8 m/s²)
F_gravity = 80 kg * 9.8 m/s²
F_gravity = 784 N
So, the gravitational force acting on the rider is 784 Newtons.
Step 2: Determine the speed required at the top:
To go around the loop, the rider must have a sufficient speed at the top to counteract the gravitational force. At the top of the loop, the gravitational force and the centripetal force are equal:
F_gravity = m * v^2 / r
where:
v = speed of the rider (15 m/s)
r = radius of the loop (18 m)
Rearranging the equation to solve for v:
v^2 = F_gravity * r / m
v^2 = 784 N * 18 m / 80 kg
v^2 = 1764 Nm / 80 kg
v^2 = 22.05 m²/s²
v ≈ √(22.05 m²/s²)
v ≈ 4.7 m/s
So, the required speed at the top of the loop is approximately 4.7 m/s.
Step 3: Calculate the net force and the force exerted by the track:
At the top of the loop, the net force is the difference between the gravitational force and the centripetal force:
F_net = F_gravity - F_centripetal
Since F_centripetal is the force exerted by the track at the top of the loop, we can rewrite it as:
F_centripetal = F_gravity - F_net
At the top of the loop, the net force is equal to zero (F_net = 0) because the rider is traveling at the minimum required speed to stay in circular motion. Therefore:
F_centripetal = F_gravity - 0
F_centripetal = F_gravity
The force exerted by the track at the top of the loop is equal to the gravitational force acting on the rider:
F_centripetal = 784 N
So, the force that the track exerts on the rider at the top of the loop is 784 Newtons.