A car while travelling at a speed of 72 km/hr. Passes through a curved portion of road in the form of an arc of a radius 10 m. If the mass of the car is 500 kg the reaction on the car at the lowest point P is-

Options

A. 25 KN
B. 50 KN
C. 75 KN
D. None of these

Answer

To solve the problem, we need to find the normal reaction force on the car at the lowest point of the curved road. We will use the principles of circular motion and Newton's laws.

### Step-by-Step Solution:

1. **Convert Speed from km/hr to m/s**:
The speed of the car is given as 72 km/hr. To convert this to meters per second (m/s), we use the conversion factor:
\[ \text{Speed in m/s} = \text{Speed in km/hr} \times \frac{5}{18} \]
\[ V = 72 \times \frac{5}{18} = 20 \text{ m/s} \]

2. **Identify the Forces Acting on the Car**:
At the lowest point of the arc, two forces act on the car:
- The gravitational force (weight) acting downwards, \( F_g = mg \)
- The normal reaction force \( N \) acting upwards

The net force providing the centripetal acceleration is the difference between the normal force and the weight of the car:
\[ N - mg = \frac{mv^2}{r} \]

3. **Calculate the Weight of the Car**:
Given the mass \( m = 500 \) kg and acceleration due to gravity \( g = 9.8 \, \text{m/s}^2 \):
\[ F_g = mg = 500 \times 9.8 = 4900 \text{ N} \]

4. **Calculate the Centripetal Force**:
The centripetal force required for the circular motion is given by:
\[ F_c = \frac{mv^2}{r} \]
Here, \( v = 20 \, \text{m/s} \) and \( r = 10 \, \text{m} \):
\[ F_c = \frac{500 \times (20)^2}{10} = \frac{500 \times 400}{10} = 20000 \text{ N} \]

5. **Set Up the Equation for Normal Force**:
Now, substituting the values into the equation:
\[ N - 4900 = 20000 \]
Rearranging gives:
\[ N = 20000 + 4900 = 24900 \text{ N} \]

6. **Convert the Normal Force to kN**:
To express the normal reaction in kilonewtons (kN):
\[ N = 24900 \text{ N} = 24.9 \text{ kN} \]

### Final Answer:
The normal reaction on the car at the lowest point \( P \) is approximately **24.9 kN**.

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