A car of mass `m` is moving on a level circular track of radius `R` if `mu_(s)` represents the static friction between the road and tyres of the car, the maximum speed of the car in circular motion is given by.

Options

A. `sqrt(Rg//mu_(s))`
B. `sqrt(mRg//mu_(s))`
C. `sqrt(mu_(s)Rg)`
D. `sqrt(mu_(s)mRg)`

Explanation

To solve the problem, we need to find the maximum speed of a car moving in a circular path on a level track, given the mass of the car, the radius of the track, and the coefficient of static friction between the tires and the road.

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Car:**
- The car is moving in a circular path, which means it requires a centripetal force to keep it in that path.
- The only force providing this centripetal force is the frictional force between the tires and the road.

2. **Write the Expression for Centripetal Force:**
- The centripetal force \( F_c \) required for circular motion is given by the formula:
\[ F_c = \frac{mv^2}{R} \]
where \( m \) is the mass of the car, \( v \) is the speed of the car, and \( R \) is the radius of the circular track.

3. **Determine the Maximum Frictional Force:**
- The maximum static frictional force \( F_f \) that can act on the car is given by:
\[ F_f = \mu_s mg \]
where \( \mu_s \) is the coefficient of static friction, \( m \) is the mass of the car, and \( g \) is the acceleration due to gravity.

4. **Set the Forces Equal:**
- For the car to maintain its circular motion without slipping, the maximum frictional force must equal the required centripetal force:
\[ \mu_s mg = \frac{mv^2}{R} \]

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

6. **Rearrange to Solve for \( v^2 \):**
- Multiply both sides by \( R \):
\[ v^2 = \mu_s g R \]

7. **Take the Square Root to Find Maximum Speed \( v \):**
- Finally, take the square root of both sides to find the maximum speed:
\[ v = \sqrt{\mu_s g R} \]

### Final Answer:
The maximum speed of the car in circular motion is given by:
\[ v = \sqrt{\mu_s g R} \]