The height of mercury column in a barometer in a Calcutta laboratory was recorded to be 75 cm. Calculate this pressure in SI and CGS units the following data, Specific gravity of mercury = 13.6, Density of `water = 10^3 kg/m^3, g=9.8 m/s^2` at Calcutta. Pressure `=h rho g` in usual symbols.

Explanation

To solve the problem of calculating the pressure exerted by a mercury column in a barometer, we will follow these steps:

### Step 1: Understand the given data
- Height of mercury column (h) = 75 cm
- Specific gravity of mercury (SG) = 13.6
- Density of water (ρ_water) = \(10^3 \, \text{kg/m}^3\)
- Acceleration due to gravity (g) = \(9.8 \, \text{m/s}^2\)

### Step 2: Convert height from cm to meters
To use SI units, we need to convert the height from centimeters to meters.
\[ h = 75 \, \text{cm} = 75 \times 10^{-2} \, \text{m} = 0.75 \, \text{m} \]

### Step 3: Calculate the density of mercury
The density of mercury (ρ_mercury) can be calculated using its specific gravity:
\[ \rho_{\text{mercury}} = \text{SG} \times \rho_{\text{water}} = 13.6 \times 10^3 \, \text{kg/m}^3 = 13600 \, \text{kg/m}^3 \]

### Step 4: Use the pressure formula
The formula for pressure (P) is given by:
\[ P = h \cdot \rho \cdot g \]
Substituting the values we have:
\[ P = 0.75 \, \text{m} \cdot 13600 \, \text{kg/m}^3 \cdot 9.8 \, \text{m/s}^2 \]

### Step 5: Calculate the pressure in SI units
Now, we calculate the pressure:
\[ P = 0.75 \cdot 13600 \cdot 9.8 \]
Calculating this step-by-step:
1. \(0.75 \cdot 13600 = 10200 \, \text{kg/m}^2\)
2. \(10200 \cdot 9.8 = 99960 \, \text{N/m}^2\)

Thus, the pressure in SI units is:
\[ P \approx 99960 \, \text{Pa} \, \text{(Pascals)} \]

### Step 6: Convert pressure to CGS units
To convert from SI units (Pascals) to CGS units (dynes/cm²), we use the following conversions:
- \(1 \, \text{N} = 10^5 \, \text{dynes}\)
- \(1 \, \text{m}^2 = 10^4 \, \text{cm}^2\)

Thus, converting the pressure:
\[ P = 99960 \, \text{Pa} = 99960 \, \text{N/m}^2 = 99960 \times 10^5 \, \text{dynes/cm}^2 \div 10^4 \]
\[ P = 99960 \times 10 \, \text{dynes/cm}^2 = 999600 \, \text{dynes/cm}^2 \]

### Final Answers
- Pressure in SI units: \(99960 \, \text{Pa}\)
- Pressure in CGS units: \(999600 \, \text{dynes/cm}^2\)

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