A roller-coaster car has a mass of 1100 kg when fully loaded with passengers. As the car passes over the top of a circular hill of radius 20 m, its speed is not changing. (a) At the top of the hill, what is the normal force (using the negative sign for the downward direction) FN on the car from the track if the car's speed is v = 7.8 m/s? (b) What is FN if v = 17 m/s?

Question

A roller-coaster car has a mass of 1100 kg when fully loaded with passengers. As the car passes over the top of a circular hill of radius 20 m, its speed is not changing. (a) At the top of the hill, what is the normal force (using the negative sign for the downward direction) FN on the car from the track if the car's speed is v = 7.8 m/s? (b) What is FN if v = 17 m/s?

Answer 1

normal force down=mg-mv^2/r

Answer 2

To determine the normal force acting on the roller-coaster car at the top of the hill, we need to consider the forces acting on the car at that point.

(a) At the top of the hill, the car is at the highest point of the circular path, and its speed is constant. Therefore, the car is in dynamic equilibrium, meaning that the net force acting on it is zero.

The forces acting on the car at the top of the hill are gravity (mg) acting downward and the normal force (FN) acting upward. Since the car is in equilibrium, the magnitudes of these forces must be equal:

|mg| = |FN|

The magnitude of the gravitational force is given by:

|mg| = m * g

where m is the mass of the car (1100 kg) and g is the acceleration due to gravity (approximately 9.8 m/s²).

Substituting the values, we have:

|mg| = (1100 kg) * (9.8 m/s²)

Now, to find FN, we use the value of the car's speed (v = 7.8 m/s) and the radius of the circular hill (r = 20 m).

The net force acting on the car at the top of the hill can be expressed as:

Net Force = Centripetal Force + Gravitational Force

The centripetal force is given by:

Centripetal Force = m * (v² / r)

Substituting the values, we have:

Centripetal Force = (1100 kg) * (7.8 m/s)² / 20 m

To find FN, we equate the centripetal force to the gravitational force:

m * (v² / r) = m * g

Simplifying the equation by canceling the mass, we have:

(v² / r) = g

Now, we can substitute the values into the equation:

(7.8 m/s)² / 20 m = 9.8 m/s²

Simplifying the equation, we find:

(7.8 m/s)² = 20 m * 9.8 m/s²

Now we can solve for FN by substituting the value of g into the equation:

FN = m * g

FN = (1100 kg) * (9.8 m/s²)

Hence, the normal force (FN) on the car from the track at the top of the hill when the car's speed is 7.8 m/s is approximately 10780 N (using the negative sign for the downward direction).

(b) To find FN when v = 17 m/s, we can follow the same steps as in part (a) but substitute the new value of v and solve for FN.