Question

A motor cyclist has to rotate in horizontal circles inside the cylindrical wall of inner radius 'R' metre. If the coefficient of friction between the wall and the tyres is '$\mu_s$', then the minimum speed required is ( g = acceleration due to gravity)

• A. $\sqrt{\mu_s Rg}$
• B. $\sqrt{\frac{Rg}{\mu_s}}$
• C. $\sqrt{\frac{\mu_s}{Rg}}$
• D. $\sqrt{\frac{R^2 g}{\mu_s}}$

Hint:

In a vertical cylinder, the normal force acts horizontally as the centripetal force, while friction acts vertically to oppose gravity.

Answer 1

The Correct Option is B

Step 1: Force Analysis
For the motorcyclist in a "Death Well," the normal reaction $N$ provides the centripetal force: $N = \frac{mv^2}{R}$. The frictional force $f_s$ balances the weight: $f_s = mg$.
Step 2: Condition for No Slipping
Friction $f_s \le \mu_s N$. Therefore, $mg \le \mu_s \left( \frac{mv^2}{R} \right)$.
Step 3: Solving for Speed
$g \le \frac{\mu_s v^2}{R} \Rightarrow v^2 \ge \frac{Rg}{\mu_s} \Rightarrow v_{min} = \sqrt{\frac{Rg}{\mu_s}}$.
Step 4: Conclusion
The minimum speed required is $\sqrt{Rg/\mu_s}$.
Final Answer:(B)