Water rises in a capillary tube to a height of 2.0cm. In another capillary tube whose radius is one third of it, how much the water will rise?

Options

A. 5 cm
B. 3 cm
C. 6 cm
D. 9 cm

Answer

To solve the problem of how much water will rise in a capillary tube whose radius is one third of the original tube, we can follow these steps:

### Step 1: Understand the relationship between height and radius in capillary action.
The height to which a liquid rises in a capillary tube is given by the formula:
\[ h = \frac{2T \cos \theta}{\rho r g} \]
where:
- \( h \) is the height of the liquid column,
- \( T \) is the surface tension of the liquid,
- \( \theta \) is the contact angle,
- \( \rho \) is the density of the liquid,
- \( r \) is the radius of the capillary tube,
- \( g \) is the acceleration due to gravity.

### Step 2: Establish the relationship between the heights and radii of the two tubes.
From the formula, we can see that the height \( h \) is inversely proportional to the radius \( r \). This means:
\[ h_1 r_1 = h_2 r_2 \]
where:
- \( h_1 \) and \( r_1 \) are the height and radius of the first tube,
- \( h_2 \) and \( r_2 \) are the height and radius of the second tube.

### Step 3: Substitute known values into the equation.
Given:
- \( h_1 = 2 \, \text{cm} \) (height in the first tube),
- \( r_2 = \frac{1}{3} r_1 \) (radius of the second tube is one third of the first).

We can rewrite the equation as:
\[ h_2 = \frac{h_1 \cdot r_1}{r_2} \]

### Step 4: Substitute \( r_2 \) in terms of \( r_1 \).
Since \( r_2 = \frac{1}{3} r_1 \), we can substitute this into the equation:
\[ h_2 = \frac{h_1 \cdot r_1}{\frac{1}{3} r_1} \]

### Step 5: Simplify the equation.
This simplifies to:
\[ h_2 = h_1 \cdot 3 \]
\[ h_2 = 2 \, \text{cm} \cdot 3 = 6 \, \text{cm} \]

### Conclusion:
The height to which the water will rise in the second capillary tube is **6 cm**.

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