Water rises in a capillary tube to a certain height such that the upward force due to surface tension is balanced by `75xx10^-4` newton force due to the weight of the liquid. If the surface tension of water is `6xx`10^-2` newton/metre the inner circumference of the capillary must be:

Options

A. ` 1.26 xx 10 ^(-2) m `
B. ` 0.50 xx 10 ^(-2) m`
C. ` 6.5 xx 10 ^(-2) m`
D. ` 0.125 `m

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the forces involved
In a capillary tube, the liquid rises due to the surface tension acting on the circumference of the tube. The upward force due to surface tension is balanced by the downward force due to the weight of the liquid column.

### Step 2: Write the equation for the force due to surface tension
The force due to surface tension can be expressed as:
\[ F = \text{Tension} \times \text{Circumference} \]
Where:
- \( F \) is the upward force due to surface tension.
- Tension (surface tension of water) is given as \( 6 \times 10^{-2} \) N/m.
- Circumference (C) is what we need to find.

### Step 3: Set up the equation
From the problem, we know that the upward force due to surface tension is balanced by the weight of the liquid, which is given as:
\[ F = 75 \times 10^{-4} \text{ N} \]

So we can set up the equation:
\[ 75 \times 10^{-4} = 6 \times 10^{-2} \times C \]

### Step 4: Solve for the circumference (C)
Rearranging the equation to solve for C:
\[ C = \frac{75 \times 10^{-4}}{6 \times 10^{-2}} \]

### Step 5: Perform the calculation
Calculating the right side:
\[ C = \frac{75 \times 10^{-4}}{6 \times 10^{-2}} = \frac{75}{6} \times \frac{10^{-4}}{10^{-2}} = \frac{75}{6} \times 10^{-2} \]
Calculating \( \frac{75}{6} \):
\[ \frac{75}{6} = 12.5 \]
Thus,
\[ C = 12.5 \times 10^{-2} \text{ m} = 0.125 \text{ m} \]

### Final Answer
The inner circumference of the capillary tube must be:
\[ C = 0.125 \text{ m} \]

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