A motorcyclist drives in a vertical circle inside a ‘death well’. What is the necessary condition so that he may not fall down?

Hint: As first step one could read the question well to understand the given condition well. Then you could think of the possibility of some condition to be followed to ensure the given situation to be true. Maybe you could think of the condition for the uppermost point related to the forces acting on the body.

Complete step-by-step solution:
In the question we are given a motorcyclist driving in a vertical circle inside a death well. We are supposed to find the necessary condition for the motorcyclist not falling down. This has to do with a condition at the upper most point of the death well. At the uppermost point of this death well, the normal force on the motor cyclist and his weight (motor cycle’s weight included) should be balanced by the centripetal force. Mathematically,
$N+mg=m{{a}_{centripetal}}=\dfrac{m{{v}^{2}}}{r}$
For this condition to be satisfied, we could say that, the velocity of the motorcyclist should be greater than the critical velocity at this top most point of the death well, so as for the motorcyclist to not drop down.

Note: In case if you wonder about the minimum speed that is required by the motorcyclist to carry out the vertical loop, then we could find it from the condition that the minimum velocity at the top most point will imply that the normal reaction on the motorcyclist is zero. So,
${{v}_{\min }}=\sqrt{rg}$