Question

Use the following information. A ride at an amusement park spins in a circle of radius \(r\) (in meters). The centripetal force \(F\) experienced by a passenger on the ride is modeled by the equation below, where \(t\) is the number of seconds the ride takes to complete one revolution and \(m\) is the mass (in kilograms) of the passenger. $$t=\sqrt{\frac{4 \pi^{2} m r}{F}}$$ A person whose mass is 67.5 kilograms is on a ride that is spinning in a circle at a rate of 10 seconds per revolution. The radius of the circle is 6 meters. How much centripetal force does the person experience?

Short answer

The passenger experiences a centripetal force of approximately 532 Newtons.

Step-by-step solution

Understanding the centripetal force equation

The equation for the centripetal force is given as \( t = \sqrt{\frac{4 \pi^{2} m r}{F}} \). In this equation, \(t\) is the time taken for one complete revolution, \(m\) is the mass of the person, \(r\) is the radius of the circle and \(F\) is the centripetal force.

Substitute the known values

First, let’s substitute the known values into our equation. We know that the mass \(m\) is 67.5 kilogram, the radius \(r\) is 6 meters, and the time \(t\) for one revolution is 10 seconds. After substitory we have the following equation: \(10=\sqrt{\frac{4 \pi^{2} \times 67.5 \times 6}{F}}\)

Solving for Force (F)

After rearrangement, we square both sides of the equation to get rid of the square root: \(100 = \frac{4 \pi^{2} \times 67.5 \times 6}{F}.\) Again rearrange the equation by multiplying both sides by F and dividing by 100 to solve for F. This gives: \(F = \frac{4 \pi^{2} \times 67.5 \times 6}{100}.\) When calculated, this gives approximately 532 Newtons.