If water be used to construct a barometer, what would be the height of water column at a standard atmospheric presure (76cm of mercury)?

Explanation

To find the height of the water column in a barometer at standard atmospheric pressure (76 cm of mercury), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between pressure and height of liquid columns**:
The pressure exerted by a liquid column is given by the formula:
\[ P = \rho g h \]
where \( P \) is the pressure, \( \rho \) is the density of the liquid, \( g \) is the acceleration due to gravity, and \( h \) is the height of the liquid column.

2. **Identify the known values**:
- The atmospheric pressure \( P_0 \) is given as 76 cm of mercury.
- The density of mercury \( \rho_{Hg} \) is approximately \( 13,600 \, \text{kg/m}^3 \).
- The density of water \( \rho_{w} \) is approximately \( 1,000 \, \text{kg/m}^3 \).
- Convert the height of mercury from centimeters to meters:
\[ h_{Hg} = 76 \, \text{cm} = 0.76 \, \text{m} \]

3. **Set up the equation for mercury**:
The pressure exerted by the mercury column can be expressed as:
\[ P_0 = \rho_{Hg} g h_{Hg} \]

4. **Set up the equation for water**:
The pressure exerted by the water column at the same height can be expressed as:
\[ P_0 = \rho_{w} g h_{w} \]

5. **Equate the two pressure equations**:
Since both expressions equal \( P_0 \), we can set them equal to each other:
\[ \rho_{Hg} g h_{Hg} = \rho_{w} g h_{w} \]

6. **Cancel \( g \) from both sides**:
Since \( g \) is present in both equations, it can be canceled out:
\[ \rho_{Hg} h_{Hg} = \rho_{w} h_{w} \]

7. **Solve for the height of the water column \( h_{w} \)**:
Rearranging the equation gives:
\[ h_{w} = \frac{\rho_{Hg} h_{Hg}}{\rho_{w}} \]

8. **Substitute the known values**:
Plug in the values for \( \rho_{Hg} \), \( h_{Hg} \), and \( \rho_{w} \):
\[ h_{w} = \frac{13,600 \, \text{kg/m}^3 \times 0.76 \, \text{m}}{1,000 \, \text{kg/m}^3} \]

9. **Calculate \( h_{w} \)**:
\[ h_{w} = \frac{10,336 \, \text{kg/m}^2}{1,000 \, \text{kg/m}^3} = 10.336 \, \text{m} \]
Rounding this gives:
\[ h_{w} \approx 10.34 \, \text{m} \]

### Final Answer:
The height of the water column at standard atmospheric pressure (76 cm of mercury) would be approximately **10.34 meters**.