Two cylindrical vessels fitted with pistons A and B of area of cross section `8 cm^(2)` and `320 cm^(2)` respectively. Are joined at their bottom by a tube and they are completely filled with water. When a mass of 4 kg is placed on piston A, find : (i) the pressure on piston A, (ii) the pressure on piston B and (iii) the thrust on piston B.

Answer

To solve the problem step by step, we will follow the instructions and calculations provided in the video transcript.

### Step 1: Calculate the Force on Piston A
The force exerted by the mass placed on piston A can be calculated using the formula:
\[ \text{Force} (F) = \text{mass} (m) \times \text{acceleration due to gravity} (g) \]

Given:
- Mass \( m = 4 \, \text{kg} \)
- Acceleration due to gravity \( g = 10 \, \text{m/s}^2 \)

Calculating the force:
\[ F = 4 \, \text{kg} \times 10 \, \text{m/s}^2 = 40 \, \text{N} \]

### Step 2: Calculate the Pressure on Piston A
Pressure is defined as the force applied per unit area. The formula for pressure \( P \) is:
\[ P = \frac{F}{A} \]

Given:
- Area of piston A \( A_A = 8 \, \text{cm}^2 = 8 \times 10^{-4} \, \text{m}^2 \) (conversion from cm² to m²)

Calculating the pressure on piston A:
\[ P_A = \frac{40 \, \text{N}}{8 \times 10^{-4} \, \text{m}^2} = \frac{40}{0.0008} = 50000 \, \text{Pa} = 5 \times 10^4 \, \text{Pa} \]

### Step 3: Calculate the Pressure on Piston B
According to Pascal's law, the pressure is transmitted equally in all directions in a fluid. Therefore, the pressure on piston B \( P_B \) is equal to the pressure on piston A \( P_A \):
\[ P_B = P_A = 5 \times 10^4 \, \text{Pa} \]

### Step 4: Calculate the Thrust on Piston B
Thrust is the force exerted by the pressure on an area. The formula for thrust \( T \) is:
\[ T = P \times A \]

Given:
- Area of piston B \( A_B = 320 \, \text{cm}^2 = 320 \times 10^{-4} \, \text{m}^2 \)

Calculating the thrust on piston B:
\[ T_B = P_B \times A_B = 5 \times 10^4 \, \text{Pa} \times 320 \times 10^{-4} \, \text{m}^2 \]
\[ T_B = 5 \times 10^4 \times 3.2 \times 10^{-2} = 1600 \, \text{N} \]

### Summary of Results
1. Pressure on piston A \( P_A = 5 \times 10^4 \, \text{Pa} \)
2. Pressure on piston B \( P_B = 5 \times 10^4 \, \text{Pa} \)
3. Thrust on piston B \( T_B = 1600 \, \text{N} \)