A roller-coaster car has a mass of 500 kg when fully loaded with passengers. The car passes over a hill of radius 15 m, as shown. At the top of the hill, the car has a speed of 8 m/s. What is the force of the track on the car at the top of the hill?

2.8 kn up

To find the force of the track on the car at the top of the hill, we need to consider the different forces acting on the car. At the top of the hill, the gravitational force and the centripetal force provide the net inward force necessary for the car to remain in circular motion.

1. Calculate the gravitational force:
The gravitational force is given by the equation Fg = m * g, where m is the mass of the car and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Fg = 500 kg * 9.8 m/s^2
Fg = 4900 N

2. Calculate the centripetal force:
The centripetal force required to keep the car in circular motion is given by the equation Fc = m * (v^2 / r), where v is the velocity of the car and r is the radius of the hill.
Fc = 500 kg * (8 m/s)^2 / 15 m
Fc = 500 kg * 8^2 m^2/s^2 / 15 m
Fc = 500 kg * 64 m^2/s^2 / 15 m
Fc = 2133.33 N

3. Calculate the net inward force:
Since the centripetal force and the gravitational force both act inward, their sum will be the net inward force.
Net inward force = Fc + Fg
Net inward force = 2133.33 N + 4900 N
Net inward force = 7033.33 N

Therefore, the force of the track on the car at the top of the hill is approximately 7033.33 N.

To find the force of the track on the car at the top of the hill, we can use the concept of centripetal force.

Centripetal force is the force that acts towards the center of a circular path and keeps an object moving in a curved path. In this case, the car is moving in a circular path on the hill.

The centripetal force is given by the equation:

F = (m * v^2) / r

Where:
F is the centripetal force,
m is the mass of the car,
v is the speed of the car, and
r is the radius of the circular path.

Given:
m = 500 kg
v = 8 m/s
r = 15 m

Let's substitute these values into the equation:

F = (500 kg * (8 m/s)^2) / 15 m

Calculating this gives us:

F = (500 kg * 64 m^2/s^2) / 15 m
F = 32000 kg * m/s^2 / 15 m
F = 2133.33 N

After calculating, we find that the force of the track on the car at the top of the hill is approximately 2133.33 Newtons.