How many revolutions per minute would a 18 m-diameter Ferris wheel need to make for the passengers to feel "weightless" at the topmost point?

when w^2 r= g

w= sqrt (g/r) and w is in rad/sec.

w (in rev/min)= w above * 60/2PI

To determine the number of revolutions per minute that a Ferris wheel with a diameter of 18 meters needs to make for passengers to feel "weightless" at the topmost point, we can use the concept of centripetal acceleration.

First, we need to calculate the velocity of the Ferris wheel at the topmost point, where the passengers will feel "weightless." At the highest point, the only force acting on the passengers is their weight, which provides the centripetal force necessary to keep them moving in a circular path.

The centripetal force (Fc) can be calculated using the formula: Fc = m * ac, where m represents the mass of the passengers and ac is the centripetal acceleration.

At the topmost point, the centripetal force is equal to the gravitational force (mg): Fc = mg.

Rearranging the formula gives us: ac = (mg) / m = g, where g is the acceleration due to gravity (approximately 9.8 m/s²).

The centripetal acceleration (ac) is also related to the velocity (v) and the radius (r) of the circular motion through the formula: ac = (v²) / r.

Since the radius of the Ferris wheel is half of its diameter (r = 18 m / 2 = 9 m), we can rewrite the formula as: ac = (v²) / 9.

Equating the two equations gives us: g = (v²) / 9.

Solving for v (velocity): v = √(g * 9).

Substituting the value of g (approximately 9.8 m/s²) gives us: v = √(9.8 * 9) = √88.2 ≈ 9.4 m/s.

Now that we have the velocity at the topmost point, we can calculate the number of revolutions per minute (RPM).

The circumference (C) of the Ferris wheel can be calculated using the formula: C = 2πr.

Substituting the value of r (9 m) gives us: C = 2π * 9 ≈ 56.55 m.

Since the Ferris wheel covers the circumference in one revolution, to find the RPM, we divide the velocity (in meters per second) by the circumference (in meters) and convert it to minutes: RPM = (v / C) * 60 ≈ (9.4 / 56.55) * 60 ≈ 9.91 RPM.

Therefore, a Ferris wheel with an 18 m-diameter would need to make approximately 9.91 revolutions per minute for the passengers to feel "weightless" at the topmost point.