A roller-coaster car speeds down a hill past point A where R1 = 9.8 m and then rolls up a hill past point B where R2 = 15.8 m (R1 and R2 represent the radiuses/ radii)

The car has a speed of 18.6 m/s at point A. if the track exerts a normal force on the car of 2.16 104 N at this point, what is the mass of the car? (in kg)

What is the maximum speed the car can have at point B for the gravitational force to hold it on the track? (m/s)

To solve this problem, we need to use the concept of centripetal force and gravitational force.

1) To find the mass of the car at point A, we can use the centripetal force equation:

F_centripetal = (mass) * (velocity^2 / radius)

At point A, the centripetal force is provided by the normal force, so we can equate these two forces:

F_centripetal = normal force

Substituting the given values:

2.16 * 10^4 N = mass * (18.6 m/s)^2 / 9.8 m

Simplifying the equation, we can find the mass:

mass = (2.16 * 10^4 N) * (9.8 m) / (18.6 m/s)^2

Calculating this gives us the mass of the car at point A.

2) To find the maximum speed the car can have at point B, we need to consider the gravitational force and the centrifugal force. The centrifugal force is provided by the normal force at point B, while the gravitational force is given by the car's weight.

At point B, the weight of the car (mg) should be equal to the force required to provide the centripetal force:

mg = (mass) * (velocity^2 / radius)

Rearranging the equation, we can solve for velocity:

velocity = sqrt((g * radius) / mass)

Substituting the given values:

velocity = sqrt((9.8 m/s^2) * 15.8 m / mass)

Calculating this gives us the maximum speed the car can have at point B.

So, by using these steps, we can find both the mass of the car at point A and the maximum speed it can have at point B.