A road is banked at an angle of `30^(@)` to the horizontal for negotiating a curve of radius `10 sqrt(3)` m. At what velocity will a car experience no friction while negotiating the curve? Take `g=10 ms^(-2)`

Options

A. 54 kmph
B. 75 kmph
C. 36 kmph
D. 18 kmph

Answer

To solve the problem of finding the velocity at which a car experiences no friction while negotiating a banked curve, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Angle of banking, \( \theta = 30^\circ \)
- Radius of the curve, \( r = 10\sqrt{3} \, \text{m} \)
- Acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \)

2. **Understand the Forces Acting on the Car:**
- The normal force \( N \) acts perpendicular to the surface of the road.
- The gravitational force \( mg \) acts vertically downwards.
- The car requires a centripetal force to negotiate the curve, which is provided by the horizontal component of the normal force.

3. **Resolve the Normal Force:**
- The normal force can be resolved into two components:
- \( N \cos \theta \) (vertical component)
- \( N \sin \theta \) (horizontal component)

4. **Set Up the Equations:**
- The vertical forces must balance:
\[ N \cos \theta = mg \]
- The horizontal forces provide the centripetal force:
\[ N \sin \theta = \frac{mv^2}{r} \]

5. **Divide the Two Equations:**
- Dividing the equation for the horizontal forces by the equation for the vertical forces, we get:
\[ \frac{N \sin \theta}{N \cos \theta} = \frac{\frac{mv^2}{r}}{mg} \]
- This simplifies to:
\[ \tan \theta = \frac{v^2}{rg} \]

6. **Substitute Known Values:**
- For \( \theta = 30^\circ \), \( \tan 30^\circ = \frac{1}{\sqrt{3}} \).
- Substitute \( r = 10\sqrt{3} \) and \( g = 10 \):
\[ \frac{1}{\sqrt{3}} = \frac{v^2}{(10\sqrt{3})(10)} \]

7. **Solve for \( v^2 \):**
- Rearranging gives:
\[ v^2 = \frac{10\sqrt{3} \cdot 10}{\sqrt{3}} = 100 \]

8. **Calculate \( v \):**
- Taking the square root gives:
\[ v = \sqrt{100} = 10 \, \text{m/s} \]

9. **Convert to km/h:**
- To convert \( v \) to kilometers per hour:
\[ v = 10 \, \text{m/s} \times \frac{18}{5} = 36 \, \text{km/h} \]

### Final Answer:
The velocity at which the car experiences no friction while negotiating the curve is **36 km/h**.