A cylinderical jar of cross-sectional area of `50 cm^(2)` is filled with water to a height of `20cm`. It carries a tight fitting piston of negligible mass. Calculate the pressure at the bottom of the jar when a mass of `1 kg` is placed on the piston . Ignore atmospheric pressure.

Answer

To solve the problem, we need to calculate the pressure at the bottom of the cylindrical jar when a mass of 1 kg is placed on the piston. We will follow these steps:

### Step 1: Calculate the force exerted by the piston
The force exerted by the piston can be calculated using the formula:
\[ F = m \cdot g \]
Where:
- \( m = 1 \, \text{kg} \) (mass placed on the piston)
- \( g = 9.8 \, \text{m/s}^2 \) (acceleration due to gravity)

Calculating the force:
\[ F = 1 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 9.8 \, \text{N} \]

### Step 2: Calculate the area of the piston in square meters
The area of the piston is given as \( 50 \, \text{cm}^2 \). We need to convert this to square meters:
\[ A = 50 \, \text{cm}^2 = 50 \times 10^{-4} \, \text{m}^2 = 0.005 \, \text{m}^2 \]

### Step 3: Calculate the pressure due to the piston
Pressure is defined as force per unit area:
\[ P_{\text{piston}} = \frac{F}{A} \]
Substituting the values:
\[ P_{\text{piston}} = \frac{9.8 \, \text{N}}{0.005 \, \text{m}^2} = 1960 \, \text{Pa} \]

### Step 4: Calculate the pressure due to the water column
The pressure due to the water column can be calculated using the formula:
\[ P_{\text{water}} = \rho \cdot g \cdot h \]
Where:
- \( \rho = 1000 \, \text{kg/m}^3 \) (density of water)
- \( g = 9.8 \, \text{m/s}^2 \) (acceleration due to gravity)
- \( h = 20 \, \text{cm} = 0.2 \, \text{m} \) (height of the water column)

Calculating the pressure:
\[ P_{\text{water}} = 1000 \, \text{kg/m}^3 \cdot 9.8 \, \text{m/s}^2 \cdot 0.2 \, \text{m} = 1960 \, \text{Pa} \]

### Step 5: Calculate the total pressure at the bottom of the jar
The total pressure at the bottom of the jar is the sum of the pressure due to the piston and the pressure due to the water column:
\[ P_{\text{total}} = P_{\text{piston}} + P_{\text{water}} \]
Substituting the values:
\[ P_{\text{total}} = 1960 \, \text{Pa} + 1960 \, \text{Pa} = 3920 \, \text{Pa} \]

### Final Answer
The pressure at the bottom of the jar when a mass of 1 kg is placed on the piston is:
\[ P_{\text{total}} = 3920 \, \text{Pa} \]

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