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Gravitational force is a fundamental force that attracts all objects towards each other. In the context of a roller-coaster on top of a hill, this force acts downward towards the center of the Earth. It contributes to the sensation of weight that passengers feel. The strength of the gravitational force is calculated by the formula:

• \( F_g = mg \)

Where:

• \( F_g \) is the gravitational force,
• \( m \) is the mass of the object, in this case, the loaded roller-coaster car (1300 kg), and
• \( g \) is the acceleration due to gravity, approximately 9.8 m/s² on the surface of the Earth.

Thus, the gravitational force acting on the roller-coaster car would be 12740 N, directing it downward as it passes over the hill's peak.

Centripetal force plays a crucial role in circular motion. It is the net force that causes an object to move in a circular path. For the roller-coaster car at the top of the hill, the centripetal force acts towards the center of the circle formed by the hill's path. This force is necessary to keep the car in its curved trajectory.

• It is calculated using the formula:\[ F_c = \frac{mv^2}{r} \]

In this equation:

• \( F_c \) is the centripetal force,
• \( m \) is the car's mass,
• \( v \) is its speed, and
• \( r \) is the radius of the circle (20 meters in this case).

For example, at a speed of 11 m/s, the centripetal force required is 7865 N. At 14 m/s, it increases to 12740 N, reflecting the greater speed's demand for stronger centripetal force to maintain the curve.

Newton's Second Law is fundamental to understanding the forces in action on the roller-coaster. It states that the force acting on an object is equal to the mass of that object multiplied by its acceleration:

• Defined as \( F = ma \).

In circular motion, this principle applies slightly differently, as the acceleration is directed towards the center of the circle. This is called centripetal acceleration, expressed by:

• \( a_c = \frac{v^2}{r} \).

Newton's Second Law helps us combine the gravitational force with the required centripetal force to understand the overall net force acting on the car. Hence, when computing forces at the top of the hill, we note that:

• \( F_g - F_N = F_c \).

This formula balances the gravitational force with the normal force and centripetal force required to keep the car on its path.

Normal force is the supporting force exerted by a surface, directed perpendicular to that surface. For the roller-coaster, as it crests the hill, the track exerts an upward normal force against the car.

• When moving over the top at 11 m/s, the normal force is 4875 N upwards.
• At 14 m/s, the normal force becomes zero as all gravitational force is used for centripetal force.

This condition means the car is at the threshold of losing contact with the track.
The magnitude of the normal force changes based on the car's speed because as speed increases, more gravitational force contributes to centripetal motion, potentially reducing or even nullifying the normal force if it equals the gravitational pull.