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Centripetal force is a force that keeps a body moving in a circular path and is always directed towards the center of the circle. This means whenever you're on a circular ride, like a Ferris wheel, the centripetal force is what keeps you in the circular motion by constantly changing your direction.

Mathematically, centripetal force (\( F_c \) ) is calculated using the formula:

• \[F_c = m a_c\]
• where \( m \) is the mass of the object (in kilograms)
• \( a_c \) is the centripetal acceleration

The centripetal acceleration \( a_c \) can be calculated with the formula: \(\[ a_c = \frac{v^2}{r} \]\)
Here, \( v \) is the constant speed of the motion and \( r \) is the radius of the circular path. In our exercise, the radius is 10 meters and the speed is 6.1 meters per second. After plugging these values into the formula, we find \( a_c = 3.72 \text{ meters per second squared} \). Ultimately, using the centripetal force equation, it's possible to find how much force is needed to keep the object moving in a circle.

The normal force is the support force exerted upon an object in contact with another stable object. In the case of the Ferris wheel, the normal force is what the seat exerts on the rider, and it varies depending on the position of the Ferris wheel.

At different points on a Ferris wheel, the normal force has to account for changes in gravitational pull. When you're at the top of the Ferris wheel, both gravity and the normal force act in the same direction towards the center of the circle:

• At the highest point: \( N_\text{high} = mg - m a_c \)
• This means the normal force is less than your weight because both the normal force and the gravitational force combine to provide the necessary centripetal force.
• When calculated, the normal force at the highest point is 486.4 N.

Conversely, at the bottom of the ride, the gravitational force acts downwards, and the seat's normal force acts upwards:

• At the lowest point: \( N_\text{low} = m a_c + mg \)
• Here, the normal force is greater than your weight because part of it must counteract gravity while still providing centripetal force.
• The calculated normal force at the lowest point is 1081.6 N.

Ferris wheel calculations often involve understanding how forces interact as you move in a vertical circle with a constant speed. Here, you calculate important variables such as the period, the normal forces at different points, and understand the effect of these forces.

**Determining the Period**
Period is the time it takes to complete one full rotation around the Ferris wheel. It can be derived from the speed and circumference of the circle using the formula:

• \(\[ T = \frac{C}{v} \]\)
• \( C \) being the circumference: \( C = 2\pi r = 62.83 \text{ meters} \)
• The period is roughly 10.3 seconds.

**Forces at Play**
While riding a Ferris wheel, several forces are always at play, especially depending on your location on the wheel. At the top, you feel lighter due to the additive nature of gravitational and centripetal forces acting in tandem. However, at the bottom, the forces offset, making the normal force from the seat feel stronger as it has to counteract gravity.
Performing these calculations helps not only in understanding the dynamics of motion on a Ferris wheel but also on other rides and vehicles that operate on similar principles like roller coasters.