A person wants to drive on the vertical surface of a large cylindrical wooden 'well' commonly known as 'death well' in a circus. The radius of the well is R and the coefficient of friction between the tyres of the motorcycle and the wall of the well is s. The minimum speed the motor cyclist must have in order to prevent slipping should be -

Options

A. `sqrt((gR)/(mu_(s)))`
B. `sqrt((mu_(s))/(gR))`
C. `sqrt((mu_(s)g)/R)`
D. `sqrt((R)/(mu_(s)g))`

Answer

To find the minimum speed the motorcyclist must have to prevent slipping while driving on the vertical surface of a cylindrical well, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Forces Acting on the Motorcycle**:
- The forces acting on the motorcycle include:
- The gravitational force (weight) acting downwards: \( F_g = mg \)
- The normal force \( N \) acting perpendicular to the surface of the well.
- The frictional force \( f \) acting upwards, opposing the gravitational force.

2. **Centripetal Force Requirement**:
- For the motorcycle to move in a circular path, a centripetal force is required. This force is provided by the normal force \( N \):
\[ N = \frac{mv^2}{R} \]
- Here, \( v \) is the speed of the motorcycle and \( R \) is the radius of the well.

3. **Frictional Force**:
- The maximum static frictional force that can act on the motorcycle is given by:
\[ f = \mu_s N \]
- Where \( \mu_s \) is the coefficient of static friction.

4. **Setting Up the Equation**:
- For the motorcycle to not slip, the frictional force must be equal to or greater than the gravitational force:
\[ f \geq mg \]
- Substituting the expression for friction:
\[ \mu_s N \geq mg \]
- Now substituting \( N = \frac{mv^2}{R} \):
\[ \mu_s \left(\frac{mv^2}{R}\right) \geq mg \]

5. **Canceling Mass**:
- Since \( m \) (the mass of the motorcycle) appears on both sides, we can cancel it out (assuming \( m \neq 0 \)):
\[ \mu_s \frac{v^2}{R} \geq g \]

6. **Rearranging for Minimum Speed**:
- Rearranging the inequality gives us:
\[ v^2 \geq \frac{gR}{\mu_s} \]
- Taking the square root of both sides to find the minimum speed:
\[ v \geq \sqrt{\frac{gR}{\mu_s}} \]

### Final Answer:
The minimum speed \( v \) the motorcyclist must have to prevent slipping is:
\[ v = \sqrt{\frac{gR}{\mu_s}} \]