water rises in a capillary tube to a height of 1 cm. In another capillary where the radius is one-third of it, how high will the water rise (in cm)?

Answer

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 follow these steps:

### Step-by-Step Solution:

1. **Understand the Capillary Rise Formula**:
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:
- \( h \) = height of the liquid column
- \( T \) = surface tension of the liquid
- \( \theta \) = angle of contact
- \( \rho \) = density of the liquid
- \( g \) = acceleration due to gravity
- \( r \) = radius of the capillary tube

2. **Identify Given Values**:
- For the first capillary tube:
- Height \( h_1 = 1 \) cm
- Radius \( r_1 = r \)
- For the second capillary tube:
- Radius \( r_2 = \frac{r}{3} \)
- Height \( h_2 \) = ? (This is what we need to find)

3. **Set Up the Relationship**:
Since the surface tension \( T \), angle of contact \( \theta \), density \( \rho \), and acceleration due to gravity \( g \) remain constant for both tubes, we can relate the heights and radii:
\[ h_1 r_1 = h_2 r_2 \]

4. **Substitute Known Values**:
Substitute the known values into the equation:
\[ 1 \cdot r = h_2 \cdot \left(\frac{r}{3}\right) \]

5. **Solve for \( h_2 \)**:
Rearranging the equation gives:
\[ h_2 = \frac{1 \cdot r}{\frac{r}{3}} = 1 \cdot 3 = 3 \text{ cm} \]

6. **Conclusion**:
Therefore, the height to which water will rise in the second capillary tube is:
\[ h_2 = 3 \text{ cm} \]

### Final Answer:
The water will rise to a height of **3 cm** in the capillary tube with a radius that is one-third of the original.

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