How many revolutions per minute would a 15 -diameter Ferris wheel need to make for the passengers to feel "weightless" at the topmost point?

11.4

To find the number of revolutions per minute (RPM) required for passengers to feel "weightless" at the topmost point of a Ferris wheel, we need to consider the forces acting on the passengers and the centripetal force required to counterbalance their weight.

The passengers will feel weightless at the topmost point when the centripetal force is equal to or greater than their weight. The centripetal force is given by the equation:

Fc = m * (v^2 / r)

Where:
Fc is the centripetal force
m is the mass of the passenger
v is the linear velocity of the Ferris wheel
r is the radius of the Ferris wheel

In this case, the mass of the passenger cancels out when comparing equal weights, so we can omit it from the equation.

We can rearrange the equation to solve for v:

v^2 = Fc * r

Since we want the passengers to feel weightless, Fc should be equal to their weight (mg).

v^2 = (mg) * r

Now we need to calculate the linear velocity (v) at the topmost point of the Ferris wheel. The linear velocity is given by:

v = 2 * π * r * RPM / 60

Where:
v is the linear velocity
r is the radius of the Ferris wheel
RPM is the revolutions per minute

We'll use this equation to express RPM in terms of v:

v = 2 * π * r * (RPM / 60)

Now let's substitute this value of v in the previous equation:

(2 * π * r * (RPM / 60))^2 = (mg) * r

Simplifying the equation:

4 * π^2 * r^2 * (RPM / 60)^2 = (mg) * r

We can rearrange it further to isolate RPM:

(RPM / 60)^2 = (mg) * r / (4 * π^2 * r^2)

RPM^2 = (mg * r) * 60^2 / (4 * π^2 * r^2)

Finally, solving for RPM:

RPM = sqrt((mg * r) * 60^2 / (4 * π^2 * r^2))

Now you can plug in the values into the equation: the mass of the passengers (m), acceleration due to gravity (g), and the radius of the Ferris wheel (r) to calculate the RPM needed for passengers to feel weightless at the topmost point.