7.0 Death well

It is a carnival sideshow mostly in India in which a car driver or motorcyclists run vehicle along the vertical wall.

Physics behind is that the normal force provides the necessary centripetal force and the weight is balance by the friction as shown in the figure.

Let a car of mass $m$ is running in a death well of radius $r$.

Therefore, $$\begin{equation} \begin{aligned} N = \frac{{m{v^2}}}{r}\quad ...(i) \\ f = mg\quad ...(ii) \\\end{aligned} \end{equation} $$

As the speed increases, centripetal force increases which in-turn increases the normal force and so the friction.

So, the minimum velocity required by car which prevents from an accident is,

$$N = \frac{{mv_{\min }^2}}{r}$$

At minimum velocity, limiting friction $\left( {{f_L}} \right)$ acts between the car tires and wall of death well.

$$\begin{equation} \begin{aligned} {f_L} = \mu N \\ {f_L} = \mu \frac{{mv_{\min }^2}}{r}\quad ...(iii) \\\end{aligned} \end{equation} $$

From equation $(ii)$ and $(iii)$ we get,

$$\begin{equation} \begin{aligned} \mu \frac{{mv_{\min }^2}}{r} = mg \\ {v_{\min }} = \sqrt {\frac{{rg}}{\mu }} \\\end{aligned} \end{equation} $$