A vessel, whose bottom has round holes with diameter of 0.1 mm , is filled with water. The maximum height to which the water can be filled without leakage is (S.T. of water =75 dyne/cm , g=1000 cm/s)

Options

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

Answer

To solve the problem of determining the maximum height to which water can be filled in a vessel with small holes at the bottom, we will use the concept of surface tension and hydrostatic pressure. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have a vessel with holes at the bottom, and we need to find the maximum height (H) of water that can be filled without leaking through the holes. The diameter of the holes is given as 0.1 mm.

### Step 2: Convert Units
First, we need to convert the diameter of the holes from millimeters to centimeters for consistency with the other units:
- Diameter (D) = 0.1 mm = 0.01 cm
- Radius (R) = D/2 = 0.01 cm / 2 = 0.005 cm

### Step 3: Use the Formula for Capillary Rise
The height of the liquid column that can be supported by surface tension is given by the formula:
\[ H = \frac{2 \cdot \text{Tension} \cdot \cos(\theta)}{\rho \cdot g \cdot R} \]
Where:
- Tension (Surface Tension) = 75 dyne/cm = 0.075 N/m (conversion to SI units)
- \(\theta\) = angle of contact (for water, \(\cos(0) = 1\))
- \(\rho\) = density of water = 1000 kg/m³ (which is 10³ g/cm³)
- \(g\) = acceleration due to gravity = 1000 cm/s² = 10 m/s² (conversion to SI units)
- \(R\) = radius in meters = 0.005 cm = 0.00005 m

### Step 4: Substitute Values into the Formula
Now we can substitute the values into the formula:
\[ H = \frac{2 \cdot 0.075 \cdot 1}{1000 \cdot 10 \cdot 0.00005} \]

### Step 5: Calculate the Height
Calculating the denominator:
- \(1000 \cdot 10 \cdot 0.00005 = 0.5\)

Now substituting this back into the equation:
\[ H = \frac{2 \cdot 0.075}{0.5} = \frac{0.15}{0.5} = 0.3 \text{ m} \]

### Step 6: Convert Height to Centimeters
To convert meters to centimeters:
\[ H = 0.3 \text{ m} = 30 \text{ cm} \]

### Final Answer
The maximum height to which the water can be filled without leakage is **30 cm**.

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