Answer
If the wheel rotates at a rate of 8.46 rpm, the riders should feel weightless.
Work Step by Step
At a certain velocity, the normal force from the seat will be zero and the riders will feel weightless. This happens when the centripetal force is equal to $F_g$:
$m \frac{v^2}{r} = mg$
$v = \sqrt{gr}$
$v = \sqrt{(9.80 ~m/s^2)(12.5 ~m)}$
$v = 11.1 ~m/s$
We can use the velocity to find the number of revolutions per second. Let z be the number of revolutions per second.
$z\times2\pi r = v$
$z = \frac{v}{2\pi r}$
$z = \frac{(11.1 ~m/s)}{(2)(\pi)(12.5 ~m)}$
$z = 0.141 ~rev/s$
The number of revolutions per minute is $60z$.
$60z = (60 ~s/min)(0.141 ~rev/s) = 8.46 ~rev/min$
If the wheel rotates at a rate of 8.46 rpm, the riders should feel weightless.
