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?

Explanation

To solve the problem of how high water will rise in a capillary tube with a radius that is one third of the original tube, we can use the principles of capillarity. The height to which a liquid rises in a capillary tube is given by the formula:

\[ H = \frac{2T \cos \theta}{\rho g R} \]

Where:
- \( H \) is the height of the liquid column,
- \( T \) is the surface tension of the liquid,
- \( \theta \) is the angle of contact,
- \( \rho \) is the density of the liquid,
- \( g \) is the acceleration due to gravity,
- \( R \) is the radius of the capillary tube.

### Step-by-Step Solution:

1. **Identify the Known Values**:
- From the problem, we know that in the first capillary tube (with radius \( R_1 \)), the height \( H_1 = 2 \, \text{cm} \).
- The radius of the second capillary tube \( R_2 \) is \( \frac{1}{3} R_1 \).

2. **Set Up the Relationship**:
- Since the formula for height \( H \) is inversely proportional to the radius \( R \), we can set up the relationship:
\[ H_1 R_1 = H_2 R_2 \]
Where \( H_2 \) is the height of the water in the second capillary tube.

3. **Substitute the Known Values**:
- Rearranging the equation gives:
\[ H_2 = \frac{H_1 R_1}{R_2} \]
- Substitute \( R_2 = \frac{1}{3} R_1 \):
\[ H_2 = \frac{H_1 R_1}{\frac{1}{3} R_1} = H_1 \times 3 \]

4. **Calculate the Height**:
- Now substitute \( H_1 = 2 \, \text{cm} \):
\[ H_2 = 2 \, \text{cm} \times 3 = 6 \, \text{cm} \]

5. **Conclusion**:
- Therefore, the height to which water will rise in the second capillary tube is **6 cm**.