A circular race track of radius 120 m is banked at an angle of `53^o` . The optimum speed of car to avoid wear and tear of its tyres is `(g = 10 ms^-2)`

Options

A. 40 m/s
B. 20 m/s
C. 30 m/s
D. 10 m/s

Answer

To find the optimum speed of a car on a banked circular track, we can use the relationship between the banking angle, radius of the track, and the speed of the car. The formula we will use is derived from the equilibrium of forces acting on the car.

### Step-by-Step Solution:

1. **Identify Given Values**:
- Radius of the track, \( r = 120 \, \text{m} \)
- Banking angle, \( \theta = 53^\circ \)
- Acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \)

2. **Use the Formula for Maximum Speed on a Banked Track**:
The formula for the maximum speed \( V \) of a car on a banked track is given by:
\[ V = \sqrt{r \cdot g \cdot \tan(\theta)} \]

3. **Calculate \( \tan(53^\circ) \)**:
We know from trigonometric tables that:
\[ \tan(53^\circ) \approx 1.327 \]
However, for practical calculations, we can use the approximate value:
\[ \tan(53^\circ) = \frac{4}{3} \]

4. **Substitute Values into the Formula**:
Now, substituting the values into the formula:
\[ V = \sqrt{120 \cdot 10 \cdot \tan(53^\circ)} \]
\[ V = \sqrt{120 \cdot 10 \cdot \frac{4}{3}} \]

5. **Simplify the Expression**:
\[ V = \sqrt{1200 \cdot \frac{4}{3}} = \sqrt{1600} \]

6. **Calculate the Final Speed**:
\[ V = 40 \, \text{m/s} \]

### Conclusion:
The optimum speed of the car to avoid wear and tear of its tyres is \( 40 \, \text{m/s} \).