A curved road of diameter 1.8 km is banked so that no friction is required at a speed of `30 ms ^(-1)` . What is the banking angle ?

Options

A. `tan ^(-1) (0.1)`
B. `tan ^(-1) (0.3)`
C. `tan ^(-1) (0.9)`
D. `tan ^(-1) (1.5)`

Answer

To find the banking angle of a curved road where no friction is required at a given speed, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Data**:
- Diameter of the road, \( D = 1.8 \, \text{km} = 1800 \, \text{m} \)
- Speed, \( V = 30 \, \text{m/s} \)
- Acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \) (approximation)

2. **Calculate the Radius of the Curve**:
\[ R = \frac{D}{2} = \frac{1800 \, \text{m}}{2} = 900 \, \text{m} \]

3. **Use the Formula for Banking Angle**:
The formula for the banking angle \( \theta \) without friction is given by:
\[ \tan \theta = \frac{V^2}{Rg} \]
where:
- \( V \) is the speed,
- \( R \) is the radius,
- \( g \) is the acceleration due to gravity.

4. **Substitute the Values into the Formula**:
\[ \tan \theta = \frac{(30 \, \text{m/s})^2}{900 \, \text{m} \times 10 \, \text{m/s}^2} \]
\[ \tan \theta = \frac{900}{9000} = \frac{1}{10} \]

5. **Calculate the Banking Angle**:
To find \( \theta \), take the arctangent (inverse tangent):
\[ \theta = \tan^{-1}\left(\frac{1}{10}\right) \]

6. **Final Calculation**:
Using a calculator, we find:
\[ \theta \approx 5.71^\circ \]

### Conclusion:
The banking angle \( \theta \) required for the curved road is approximately \( 5.71^\circ \).

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