In a vehicle lifter the enclosed gas exerts a force F on a small piston having a diameter of 8 cm. This pressure is transmitted to a second piston of diameter 24 cm. If the mass of the vehicle to be lifted is 1400 kg then value of F is

Options

A. 1200 N
B. 1800 N
C. 1600 N
D. 700 N

Explanation

To solve the problem of finding the force \( F \) exerted on the small piston in a vehicle lifter, we will use the principle of pressure balance. Here are the steps to arrive at the solution:

### Step 1: Understand the relationship between pressure and force
The pressure exerted by a fluid is given by the formula:
\[ P = \frac{F}{A} \]
where \( P \) is the pressure, \( F \) is the force, and \( A \) is the area of the piston.

### Step 2: Identify the areas of the pistons
The area \( A \) of a piston can be calculated using the formula for the area of a circle:
\[ A = \pi r^2 \]
where \( r \) is the radius of the piston. Given the diameters:
- Diameter of the small piston \( d_1 = 8 \, \text{cm} = 0.08 \, \text{m} \) (convert to meters)
- Diameter of the large piston \( d_2 = 24 \, \text{cm} = 0.24 \, \text{m} \)

Calculate the radii:
- \( r_1 = \frac{d_1}{2} = \frac{0.08}{2} = 0.04 \, \text{m} \)
- \( r_2 = \frac{d_2}{2} = \frac{0.24}{2} = 0.12 \, \text{m} \)

Now calculate the areas:
\[ A_1 = \pi r_1^2 = \pi (0.04)^2 = \pi (0.0016) \approx 0.0050265 \, \text{m}^2 \]
\[ A_2 = \pi r_2^2 = \pi (0.12)^2 = \pi (0.0144) \approx 0.0452389 \, \text{m}^2 \]

### Step 3: Set up the pressure balance equation
According to Pascal's principle, the pressure in the fluid is the same at both pistons:
\[ P_1 = P_2 \]
This gives us:
\[ \frac{F}{A_1} = \frac{F_2}{A_2} \]
where \( F_2 \) is the weight of the vehicle, which can be calculated as:
\[ F_2 = mg \]
Given the mass \( m = 1400 \, \text{kg} \) and using \( g \approx 9.81 \, \text{m/s}^2 \):
\[ F_2 = 1400 \times 9.81 \approx 13734 \, \text{N} \]

### Step 4: Substitute the values into the pressure equation
Substituting the values into the pressure balance equation:
\[ \frac{F}{0.0050265} = \frac{13734}{0.0452389} \]

### Step 5: Solve for \( F \)
Cross-multiplying gives:
\[ F \cdot 0.0452389 = 13734 \cdot 0.0050265 \]
Calculating the right side:
\[ F \cdot 0.0452389 = 68.96 \]
Now, solving for \( F \):
\[ F = \frac{68.96}{0.0452389} \approx 1526.5 \, \text{N} \]

### Step 6: Finalize the result
Rounding to the nearest whole number, we find:
\[ F \approx 1527 \, \text{N} \]

### Conclusion
The value of the force \( F \) exerted on the small piston is approximately **1527 N**.

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