The general equation for the forces acting on the passengers at the topmost point of the ferris wheel is
[tex]mg - R = m \omega^2 r[/tex]
where
mg is the weight of the passengers
R is the normal reaction of the cabin
[tex]m \omega^2 r[/tex] is the centripetal force
In order to feel weightless, the normal reaction felt by the passengers should be zero. Therefore, the equation becomes:
[tex]mg=m \omega^2 r[/tex]
or
[tex]g=\omega^2 r[/tex]
where [tex]\omega[/tex] is the angular frequency of the wheel and r is its radius. Since we know its radius,
[tex]r= \frac{20 m}{2}=10 m [/tex]
we can calculate the angular frequency:
[tex]\omega= \sqrt{ \frac{g}{r} } = \sqrt{ \frac{9.81 m/s^2}{10 m} } =0.99 rad/s[/tex]
From which we find the frequency at which the ferris wheel should rotate:
[tex]f= \frac{\omega}{2 \pi}= \frac{0.99 rad/s}{2 \pi}=0.158 s^{-1} [/tex]
This is the number of revolutions per second, so the number of revolutions per minute will be
[tex]f=0.158 s^{-1} \cdot 60 = 9.48 min^{-1}[/tex]
