You may have seen in a circus a motorcyclist driving in vertical loops inside a 'death well' (a hollow spherical chamber with holes,so the spectators can watch from outside). Explain clearly why the motorcyclist does not drop down when he is at the uppermost point,with no support from below. What is the minimum speed required at the uppermost position to perform a vertical loop if the radius of the chamber is $25 \; m$?

(N/A) In a death well,a motorcyclist does not fall at the top point of a vertical loop because the gravitational force and the normal reaction force from the wall act downward,providing the necessary centripetal force for circular motion.
The net force acting on the motorcyclist at the top is the sum of the normal force $(F_N)$ and the force due to gravity $(F_g = mg)$.
The equation of motion for the centripetal acceleration $(a_c)$ is:
$F_{\text{net}} = m a_c$
$F_N + mg = \frac{m v^2}{r}$
For the motorcyclist to just complete the loop without falling,the minimum speed $(v_{\min})$ occurs when the normal reaction $(F_N)$ becomes zero.
$mg = \frac{m v_{\min}^2}{r}$
$v_{\min}^2 = rg$
$v_{\min} = \sqrt{rg}$
Given $r = 25 \; m$ and taking $g = 10 \; m/s^2$:
$v_{\min} = \sqrt{25 \times 10} = \sqrt{250} \approx 15.81 \; m/s$.