A car is moving on a circular level road of curvature `300m` . If the coefficient of friction is `0.3` and acceleration due to gravity is `10m//s^(2)` , the maximum speed of the car be a) 90 km/h b) 81 km/h c) 108 km/h d) 162 km/h

Options

A. `90Km//h`
B. `81Km//h`
C. `108Km//h`
D. `162Km//h`

Explanation

To find the maximum speed of the car moving on a circular level road, we can use the relationship between frictional force, centripetal force, and the parameters given in the problem. Here’s a step-by-step solution:

### Step 1: Identify the given values
- Radius of curvature (r) = 300 m
- Coefficient of friction (μ) = 0.3
- Acceleration due to gravity (g) = 10 m/s²

### Step 2: Write the equation for centripetal force
The centripetal force required to keep the car moving in a circular path is given by:
\[ F_c = \frac{mv^2}{r} \]
where:
- \( m \) is the mass of the car,
- \( v \) is the speed of the car,
- \( r \) is the radius of the circular path.

### Step 3: Write the equation for frictional force
The maximum frictional force that can act on the car is given by:
\[ F_f = \mu N \]
where:
- \( N \) is the normal force. For a level road, \( N = mg \), so:
\[ F_f = \mu mg \]

### Step 4: Set the centripetal force equal to the frictional force
For the car to move without slipping, the maximum frictional force must equal the required centripetal force:
\[ \mu mg = \frac{mv^2}{r} \]

### Step 5: Cancel the mass (m) from both sides
Since mass \( m \) appears on both sides of the equation, we can cancel it out:
\[ \mu g = \frac{v^2}{r} \]

### Step 6: Rearrange the equation to solve for \( v^2 \)
Rearranging gives us:
\[ v^2 = \mu g r \]

### Step 7: Substitute the known values into the equation
Now we can substitute the values of \( \mu \), \( g \), and \( r \):
\[ v^2 = 0.3 \times 10 \, \text{m/s}^2 \times 300 \, \text{m} \]
\[ v^2 = 0.3 \times 10 \times 300 = 900 \]

### Step 8: Take the square root to find \( v \)
Taking the square root gives us:
\[ v = \sqrt{900} = 30 \, \text{m/s} \]

### Step 9: Convert the speed from m/s to km/h
To convert from meters per second to kilometers per hour, we use the conversion factor:
\[ 1 \, \text{m/s} = 3.6 \, \text{km/h} \]
Thus:
\[ v = 30 \, \text{m/s} \times 3.6 = 108 \, \text{km/h} \]

### Final Answer
The maximum speed of the car is **108 km/h**, which corresponds to option (c).

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