The diameter of the piston of a hydraulic automobile is D metre. What pressure, in atmosphere is required to lift a car of mass m kg ?

Options

A. `(4mg)/(pi D^2 xx 10^5)`
B. `(2mg)/(pi D^2)`
C. `(mg)/(pi D^2)`
D. `10^5 xx (4mg)/(pi D^2)`

Explanation

To find the pressure required to lift a car of mass \( m \) kg using a hydraulic piston with a diameter \( D \) meters, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definition of Pressure**:
Pressure (\( P \)) is defined as the force (\( F \)) acting per unit area (\( A \)):
\[ P = \frac{F}{A} \]

2. **Identify the Force Acting on the Piston**:
The force acting on the piston due to the weight of the car is given by:
\[ F = mg \]
where \( m \) is the mass of the car and \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)).

3. **Calculate the Area of the Piston**:
The area (\( A \)) of the piston can be calculated using the formula for the area of a circle:
\[ A = \frac{\pi}{4} D^2 \]
where \( D \) is the diameter of the piston.

4. **Substitute the Values into the Pressure Formula**:
Now, substituting the expressions for force and area into the pressure formula:
\[ P = \frac{mg}{\frac{\pi}{4} D^2} = \frac{4mg}{\pi D^2} \]

5. **Convert Pressure from Pascals to Atmospheres**:
We know that \( 1 \, \text{atmosphere} = 10^5 \, \text{Pascals} \). Therefore, to convert the pressure from Pascals to atmospheres, we divide by \( 10^5 \):
\[ P_{\text{atm}} = \frac{P}{10^5} = \frac{4mg}{\pi D^2 \cdot 10^5} \]

### Final Answer:
The pressure required to lift the car in atmospheres is:
\[ P_{\text{atm}} = \frac{4mg}{\pi D^2 \cdot 10^5} \]