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. 6.0 cm
B. 2.0 cm
C. 4.0 cm
D. 8.0 cm

Explanation

To solve the problem of how high water will rise in a capillary tube with a radius one-third that of another tube where water rises to a height of 2.0 cm, we can use the relationship between the height of the liquid column and the radius of the capillary tube.

### Step-by-Step Solution:

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

2. **Identify the Given Values**:
- In the first capillary tube (let's call it Tube 1), the height \( h_1 = 2.0 \, \text{cm} \).
- The radius of the second tube (Tube 2) is one-third of Tube 1, so if \( r_1 \) is the radius of Tube 1, then \( r_2 = \frac{r_1}{3} \).

3. **Relate the Heights and Radii**:
From the capillary rise formula, we can derive a relationship between the heights and radii of the two tubes:
\[ h_1 r_1 = h_2 r_2 \]
Rearranging gives us:
\[ h_2 = \frac{h_1 r_1}{r_2} \]

4. **Substituting the Values**:
Now substitute \( r_2 = \frac{r_1}{3} \) into the equation:
\[ h_2 = \frac{h_1 r_1}{\frac{r_1}{3}} = h_1 \cdot 3 \]
Since \( h_1 = 2.0 \, \text{cm} \):
\[ h_2 = 2.0 \, \text{cm} \cdot 3 = 6.0 \, \text{cm} \]

5. **Final Result**:
Therefore, the height to which water will rise in the second capillary tube is:
\[ h_2 = 6.0 \, \text{cm} \]