When the road is dry and coefficient of friction is `mu`. the maximum speed of a cur in a circular path is `10m//sec`, If the road becomes wet and `mu^(1) = mu//2`. What is the maximum speed permitted

Options

A. `5m//sec`
B. `10 m// sec`
C. `10 sqrt(2) m//sec`
D. `5sqrt(2) m//sec`

Answer

To solve the problem step by step, we need to analyze the forces acting on a car moving in a circular path and how they change when the coefficient of friction changes due to wet conditions.

### Step-by-Step Solution:

1. **Understanding the Forces**:
When a car is moving in a circular path, the frictional force provides the necessary centripetal force to keep the car in motion. The maximum frictional force can be expressed as:
\[ f_{\text{max}} = \mu \cdot N \]
where \( N \) is the normal force, which in this case is equal to the weight of the car \( mg \).

2. **Setting Up the Equation for Dry Conditions**:
For the dry road, the maximum speed \( v \) of the car is given as \( 10 \, \text{m/s} \). The centripetal force required for circular motion can be expressed as:
\[ f_{\text{max}} = \frac{mv^2}{r} \]
Setting the frictional force equal to the centripetal force, we have:
\[ \mu mg = \frac{mv^2}{r} \]
Here, the mass \( m \) cancels out, leading to:
\[ \mu g = \frac{v^2}{r} \]

3. **Calculating for Wet Conditions**:
When the road becomes wet, the coefficient of friction changes to \( \mu' = \frac{\mu}{2} \). We need to find the new maximum speed \( v' \):
\[ \mu' mg = \frac{mv'^2}{r} \]
Again, canceling \( m \) gives:
\[ \frac{\mu}{2} g = \frac{v'^2}{r} \]

4. **Relating the Two Conditions**:
From the dry condition, we have:
\[ \mu g = \frac{10^2}{r} \quad \text{(since } v = 10 \text{ m/s)} \]
Substituting this into the wet condition:
\[ \frac{\mu}{2} g = \frac{v'^2}{r} \]
This implies:
\[ \frac{1}{2} \left(\frac{10^2}{r}\right) = \frac{v'^2}{r} \]
Thus, we can simplify to:
\[ \frac{100}{2} = v'^2 \]
Therefore:
\[ v'^2 = 50 \]

5. **Finding the Maximum Speed**:
Taking the square root gives:
\[ v' = \sqrt{50} = 5\sqrt{2} \, \text{m/s} \]

### Final Answer:
The maximum speed permitted when the road is wet is \( 5\sqrt{2} \, \text{m/s} \).