Riders on the "Great American Revolution" loop-the-loop roller coaster of Example 5.7 wear seatbelts as the roller coaster negotiates its 6.3 -m-rudius loop at \(9.7 \mathrm{m} / \mathrm{s} .\) At the top of the loop. what are the magnitude and direction of the force exerted on a 60-kg rider (a) by the roller-coaster seat and (b) by the seatbelt? (c) What would happen if the rider unbuckled at this point?

Understanding the gravitational force on a roller coaster is essential for ensuring rider safety and the thrill of the experience. Gravitational force, denoted as \(F_g\), influences everything on Earth and, of course, those riding roller coasters. It’s calculated by the equation \(F_g = m \cdot g\),
where \(m\) represents the mass of the object and \(g\) is the acceleration due to gravity, which is approximately \(9.8 \, m/s^2\) on Earth's surface.

For a rider with a mass of \(60 \, kg\), just like in our exercise, the gravitational force would be calculated as follows: \(F_g = 60 \, kg \cdot 9.8 \, m/s^2 = 588 \, N\). This force acts downwards towards the center of the Earth. In roller coaster design, this force contributes to the thrilling sensation of weightlessness and the heavy feeling of increased G-forces.

Roller coasters are the perfect examples of physics in action, showcasing how centripetal force keeps riders in a curved path. Centripetal force, \(F_c\), is the force that keeps an object moving in a circular path and is directed towards the center of the circle.
The equation for centripetal force is \(F_c = \frac{m \cdot v^2}{r}\),
where \(m\) is the mass of the object, \(v\) is the velocity at which the object moves, and \(r\) is the radius of the circular path. For our roller coaster rider, calculating the centripetal force at the top of a 6.3-meter radius loop with a velocity of \(9.7 \, m/s\) would give us \(F_c = \frac{60 \, kg\cdot (9.7 \, m/s)^2}{6.3 \, m} ≈ 882 \, N\).

This inward force is crucial for the roller coaster design, ensuring riders stay securely on their path throughout loops and turns. Understanding this concept allows designers to create loops that are thrilling yet safe, as the centripetal force must be sufficient to overcome the gravitational force at the top of the loop.

Normal force is another key player when it comes to understanding the physics of roller coasters, especially in circular motion like loop-the-loops. It is the force exerted by a surface perpendicular to the object in contact with it. At the top of the loop, the normal force \(F_N\) works against gravity to provide the necessary centripetal force.

The calculation for the normal force in our exercise is found by subtracting the gravitational force from the required centripetal force: \(F_N = F_c - F_g = 882 \, N - 588 \, N = 294 \, N\). This tells us that the roller coaster seat must provide an upward force of 294 N to keep the rider in circular motion. Without it, such as if the rider were to unbuckle their seatbelt, they would not stay securely in their seat due to the lack of counteracting force against gravity, highlighting the importance of safety restraints on such rides.