A long capillary tube of radius `0.2 mm` is placed vertically inside a beaker of water.
If the surface tension of water is `7.2xx10^(-2)N//m` the angle of contact between glass and water is zero, then determine the height of the water column in the tube.

Options

A. `3cm`
B. `9cm`
C. `7cm`
D. `5cm`

Explanation

To determine the height of the water column in a capillary tube, we can use the capillary rise formula:

\[ h = \frac{2S \cos \theta}{r \rho g} \]

Where:
- \( h \) = height of the water column
- \( S \) = surface tension of the liquid (water in this case)
- \( \theta \) = angle of contact (given as 0 degrees)
- \( r \) = radius of the capillary tube
- \( \rho \) = density of the liquid (water)
- \( g \) = acceleration due to gravity

### Step 1: Gather the given values
- Radius of the capillary tube, \( r = 0.2 \, \text{mm} = 0.2 \times 10^{-3} \, \text{m} \)
- Surface tension of water, \( S = 7.2 \times 10^{-2} \, \text{N/m} \)
- Angle of contact, \( \theta = 0 \, \text{degrees} \) (thus, \( \cos \theta = 1 \))
- Density of water, \( \rho = 10^{3} \, \text{kg/m}^3 \)
- Acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \)

### Step 2: Substitute the values into the formula
Using the formula for height \( h \):

\[ h = \frac{2S \cos \theta}{r \rho g} \]

Substituting the known values:

\[ h = \frac{2 \times (7.2 \times 10^{-2}) \times 1}{(0.2 \times 10^{-3}) \times (10^{3}) \times (10)} \]

### Step 3: Simplify the expression
First, calculate the denominator:

\[ (0.2 \times 10^{-3}) \times (10^{3}) \times (10) = 0.2 \times 10 = 2 \]

Now substitute this back into the equation for \( h \):

\[ h = \frac{2 \times (7.2 \times 10^{-2})}{2} \]

The 2s in the numerator and denominator cancel out:

\[ h = 7.2 \times 10^{-2} \, \text{m} \]

### Step 4: Convert to centimeters
To convert meters to centimeters:

\[ h = 7.2 \times 10^{-2} \, \text{m} = 7.2 \, \text{cm} \]

### Step 5: Round the answer
Since the problem asks for an approximate value, we can round it:

\[ h \approx 7 \, \text{cm} \]

### Final Answer
The height of the water column in the tube is approximately \( 7 \, \text{cm} \).

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