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Chapter 5: Problem 60

A person of mass \(M\) stands on a bathroom scale inside a Ferris wheel compartment. The Ferris wheel has radius \(R\) and angular velocity \(\omega\). What is the apparent weight of the person (a) at the top and (b) at the bottom?

Short Answer

Expert verified
Question: Calculate the apparent weight of a person with mass M standing on a bathroom scale inside a Ferris wheel compartment with radius R and angular velocity 蠅. Find the person's apparent weight at the (a) top and (b) bottom of the Ferris wheel. Answer: (a) At the top, the person's apparent weight is \(W_t = M\cdot g - M\cdot (R\omega^2)\). (b) At the bottom, the person's apparent weight is \(W_b = M\cdot g + M\cdot (R\omega^2)\).

Step by step solution

01

Write the equation for centripetal force

The centripetal force acting on the person is determined by: \(F_c = M\cdot(R\omega^2)\) Where \(F_c\) is the centripetal force, \(M\) is the mass of the person, \(R\) is the radius of the Ferris wheel, and \(\omega\) is the angular velocity.
02

Write the equation for gravitational force

The gravitational force acting on the person is given by: \(F_g = M\cdot g\) Where \(F_g\) is the gravitational force, \(M\) is the mass of the person, and \(g\) is the acceleration due to gravity (approximately \(9.8 ms^{-2}\)).
03

Find the apparent weight at the top of the Ferris wheel

When the person is at the top, the centripetal force acts downward in the same direction as the gravitational force. The apparent weight will be the difference between the gravitational force and the centripetal force. Therefore, the person's apparent weight \(W_t\) at the top is: \(W_t = F_g - F_c = M\cdot g - M\cdot (R\omega^2)\)
04

Find the apparent weight at the bottom of the Ferris wheel

When the person is at the bottom, the centripetal force acts upward, opposite to the gravitational force. The apparent weight will be the sum of the gravitational force and the centripetal force. Therefore, the person's apparent weight \(W_b\) at the bottom is: \(W_b = F_g + F_c = M\cdot g + M\cdot (R\omega^2)\)
05

Write the results

So, the person's apparent weight on the Ferris wheel is: (a) At the top: \(W_t = M\cdot g - M\cdot (R\omega^2)\) (b) At the bottom: \(W_b = M\cdot g + M\cdot (R\omega^2)\)

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