The Cosmo Clock 21 Ferris wheel in Yokohama, Japan, has a diameter of 100 m. Its name comes from its 60 arms, each of which can function as a second hand (so that it makes one revolution every 60.0 s).(c) What would be the time for one revolution if the passenger's apparent weight at the highest point were zero?

Verified step by step guidance

1. Understand that the apparent weight of a passenger at the highest point of the Ferris wheel being zero implies that the passenger is in a state of free fall at that point. This means the only force acting on the passenger is gravity, which provides the necessary centripetal force for the circular motion of the Ferris wheel.
2. Recall the formula for centripetal force, which is given by \( F_c = \frac{m v^2}{r} \), where \( m \) is the mass of the object, \( v \) is the velocity, and \( r \) is the radius of the circular path.
3. Set up the equation for the condition where the apparent weight is zero. The gravitational force (mg) will equal the centripetal force needed to keep the passenger moving in a circle: \( mg = \frac{m v^2}{r} \).
4. Simplify the equation to find the velocity \( v \) at the top of the Ferris wheel where \( v = \sqrt{g \cdot r} \), with \( g \) being the acceleration due to gravity (approximately 9.8 m/s^2) and \( r \) being half of the diameter of the Ferris wheel.
5. Calculate the time for one revolution based on this new velocity. The circumference of the Ferris wheel is \( 2\pi r \). The time for one revolution (period T) can be found using \( T = \frac{2\pi r}{v} \).

Key Concepts

Centripetal Force

Centripetal force is the net force acting on an object moving in a circular path, directed towards the center of the circle. In the context of the Ferris wheel, this force is necessary to keep passengers moving in a circular motion. At the highest point of the ride, the gravitational force and the normal force together provide the required centripetal force.

Apparent Weight

Apparent weight refers to the sensation of weight experienced by an object or person, which can differ from actual weight due to acceleration. In this scenario, when the passenger's apparent weight is zero at the highest point, it indicates that the normal force acting on them is zero, meaning that the gravitational force alone provides the necessary centripetal force for circular motion.

Period of Revolution

The period of revolution is the time taken for one complete cycle of motion, such as a full rotation of the Ferris wheel. It is influenced by the radius of the circular path and the acceleration due to gravity. To find the period when the apparent weight is zero, one must consider the balance of forces acting on the passenger at the highest point, which affects the time it takes to complete a revolution.