A car is travelling at 36 kmph on a road. If `mu=0.5` between the tyres and the road, the minimum turning radius of the car is: (`g = 10 ms^(-2)`)

Options

A. 20 m
B. 25 m
C. 30 m
D. 35 m

Answer

To solve the problem of finding the minimum turning radius of a car traveling at 36 km/h with a coefficient of friction (μ) of 0.5, we can follow these steps:

### Step 1: Convert the speed from km/h to m/s
The speed of the car is given as 36 km/h. We need to convert this to meters per second (m/s) using the conversion factor \(1 \text{ km/h} = \frac{1}{3.6} \text{ m/s}\).

\[ \text{Speed} = 36 \text{ km/h} \times \frac{1 \text{ m/s}}{3.6 \text{ km/h}} = 10 \text{ m/s} \]

### Step 2: Identify the forces acting on the car
When the car is turning, the frictional force provides the necessary centripetal force to keep the car moving in a circular path. The frictional force (F_f) can be expressed as:

\[ F_f = \mu \cdot m \cdot g \]

Where:
- \( \mu = 0.5 \) (coefficient of friction)
- \( m \) is the mass of the car
- \( g = 10 \text{ m/s}^2 \) (acceleration due to gravity)

### Step 3: Write the expression for centripetal force
The centripetal force (F_c) required to keep the car moving in a circular path is given by:

\[ F_c = \frac{m \cdot v^2}{r} \]

Where:
- \( v = 10 \text{ m/s} \) (velocity of the car)
- \( r \) is the turning radius we want to find.

### Step 4: Set the frictional force equal to the centripetal force
For the car to turn without skidding, the frictional force must equal the centripetal force:

\[ \mu \cdot m \cdot g = \frac{m \cdot v^2}{r} \]

### Step 5: Simplify the equation
We can cancel the mass \( m \) from both sides of the equation:

\[ \mu \cdot g = \frac{v^2}{r} \]

### Step 6: Solve for the turning radius \( r \)
Rearranging the equation to solve for \( r \):

\[ r = \frac{v^2}{\mu \cdot g} \]

### Step 7: Substitute the known values
Now, we substitute \( v = 10 \text{ m/s} \), \( \mu = 0.5 \), and \( g = 10 \text{ m/s}^2 \):

\[ r = \frac{(10)^2}{0.5 \cdot 10} = \frac{100}{5} = 20 \text{ m} \]

### Conclusion
The minimum turning radius of the car is \( r = 20 \text{ m} \).

---