Question

A roller coaster has a "hump" and a "loop" for riders to enjoy (see picture). The top of the hump has a radius of curvature of 12 m and the loop has a radius of curvature of 15 m. (a) When going over the hump, the coaster is traveling with a speed of 9.0 m/s. A 100-kg rider is traveling on the coaster. What is the normal force of the rider’s seat on the rider when he is at the peak of the hump? Compare this with the normal force he would experience when the coaster is at rest. (b) What is the minimum speed the coaster must have at the top of the loop in order for the rider to remain in contact with his seat? Is this speed dependent on the mass of the rider?

Answer 1

To find the normal force of the rider's seat on the rider at the peak of the hump, we need to consider the forces acting on the rider. At the top of the loop, the net force acting on the rider is the centripetal force. To find the minimum speed the coaster must have at the top of the loop in order for the rider to remain in contact with their seat, we need to consider the forces acting on the rider. The speed at the top of the loop does not depend on the mass of the rider.

Step-by-step explanation:

(a) To find the normal force of the rider's seat on the rider at the peak of the hump, we need to consider the forces acting on the rider. At the top of the hump, the net force acting on the rider is the centripetal force. The centripetal force is provided by the normal force and the gravitational force. At the peak of the hump, the rider experiences the maximum normal force, which is equal to the sum of their weight and the centripetal force. The normal force can be calculated using the formula:

Normal Force = Weight + Centripetal Force

where Weight = mass x gravitational acceleration and Centripetal Force = mass x (velocity squared / radius).

Substituting the given values into the formula, we can calculate the normal force. To compare the normal force the rider would experience when the coaster is at rest, we can calculate the normal force using the formula Normal Force = mass x gravitational acceleration.

(b) To find the minimum speed the coaster must have at the top of the loop in order for the rider to remain in contact with their seat, we need to consider the forces acting on the rider. At the top of the loop, the net force acting on the rider is the centripetal force. The centripetal force is provided by the normal force and the gravitational force. To remain in contact with their seat, the rider must experience a minimum normal force equal to their weight. The minimum speed can be calculated using the formula for centripetal force: Centripetal Force = mass x (velocity squared / radius). By substituting the given values into the formula and solving for velocity, we can find the minimum speed required. The speed at the top of the loop does not depend on the mass of the rider, as it cancels out in the calculation of the centripetal force.

Answer 2

Part a)

`F_n = 306 N`

Part b)

`v = 12.1 m/s`

So this speed is independent of the mass of the rider

Step-by-step explanation:

Part a)

By force equation on the rider at the position of the hump we can say

`mg - F_n = ma_c`

now we will have

`mg - F_n = (mv^2)/(R)`

`F_n = mg - (mv^2)/(R)`

now we have

`F_n = 100(9.81) - (100(9^2))/(12)`

`F_n = 981 - 675`

`F_n = 306 N`

Part b)

At the top of the loop if the minimum speed is required so that it remains in contact so we will have

`F_n + mg = ma_c`

`F_n = 0` at minimum speed

`mg = (mv^2)/(R)`

`v = √(Rg)`

`v = √(15 * 9.81)`

`v = 12.1 m/s`

So this speed is independent of the mass of the rider