A vessel whose , bottom has round holes with diameter `0.1 mm`, is filled with water. The maximum height up to which water can be filled without leakage is

Options

A. `100 cm`
B. `75 cm`
C. `50 cm`
D. `30 cm`

Explanation

To find the maximum height up to which water can be filled in a vessel with a hole at the bottom, we can use the principles of fluid mechanics, specifically the relationship between pressure, surface tension, and the geometry of the hole.

### Step-by-Step Solution:

1. **Understand the Problem**:
We have a vessel with a hole at the bottom, and we need to determine the maximum height of water that can be filled without leaking out through the hole.

2. **Identify Given Data**:
- Diameter of the hole, \(d = 0.1 \, \text{mm} = 0.1 \times 10^{-3} \, \text{m} = 1 \times 10^{-4} \, \text{m}\)
- Radius of the hole, \(R = \frac{d}{2} = \frac{1 \times 10^{-4}}{2} = 0.5 \times 10^{-4} \, \text{m}\)
- Density of water, \(\rho \approx 1000 \, \text{kg/m}^3\)
- Acceleration due to gravity, \(g \approx 9.8 \, \text{m/s}^2\)
- Surface tension of water, \(T \approx 0.072 \, \text{N/m}\)

3. **Apply the Pressure Balance**:
The pressure at the bottom of the vessel due to the height of the water column must balance the pressure drop due to surface tension at the hole. The pressure at the bottom is given by:
\[ P = P_{\text{atm}} + \rho g h \]
The pressure drop due to surface tension at the hole can be expressed as:
\[ \Delta P = \frac{2T}{R} \]
Setting these equal gives:
\[ \rho g h = \frac{2T}{R} \]

4. **Rearranging for Height**:
We can rearrange the equation to solve for \(h\):
\[ h = \frac{2T}{\rho g R} \]

5. **Substituting Values**:
Substitute the known values into the equation:
\[ h = \frac{2 \times 0.072}{1000 \times 9.8 \times (0.5 \times 10^{-4})} \]

6. **Calculating**:
First calculate the denominator:
\[ 1000 \times 9.8 \times (0.5 \times 10^{-4}) = 1000 \times 9.8 \times 0.000005 = 0.049 \]
Now substitute back:
\[ h = \frac{0.144}{0.049} \approx 2.94 \, \text{m} \]

7. **Final Result**:
The maximum height up to which water can be filled without leakage is approximately \(2.94 \, \text{m}\).