A capillary tube of the radius r is immersed in water and water rise in it to a height H. Mass of water in the capillary tube is m. If the capillary of radius 2r is taken and dipped in water, the mass of water that will rise in the capillary tube will be

Options

A. m
B. 2 m
C. `m/2`
D. 4 m

Answer

To solve the problem, we need to understand the relationship between the radius of the capillary tube and the height to which the liquid rises due to capillary action. The key principle we will use is that the product of the radius of the capillary tube and the height of the liquid column is constant.

### Step-by-Step Solution:

1. **Understand the Initial Conditions**:
- Let the radius of the first capillary tube be \( r \).
- The height of the water that rises in this tube is \( H \).
- The mass of water in this tube is \( m \).

2. **Apply the Capillary Rise Formula**:
- The relationship for capillary rise is given by:
\[ r_1 \cdot h_1 = r_2 \cdot h_2 \]
- Here, \( r_1 = r \), \( h_1 = H \), and for the second capillary tube, \( r_2 = 2r \) and we need to find \( h_2 \).

3. **Calculate the Height for the Second Capillary**:
- Substituting the values into the formula:
\[ r \cdot H = (2r) \cdot h_2 \]
- Simplifying this gives:
\[ H = 2h_2 \implies h_2 = \frac{H}{2} \]

4. **Calculate the Volume of Water in the Second Capillary**:
- The volume of water that rises in the second capillary tube can be calculated using the formula for the volume of a cylinder:
\[ V = \pi r^2 h \]
- For the second capillary tube, the radius is \( 2r \) and the height is \( h_2 = \frac{H}{2} \):
\[ V_2 = \pi (2r)^2 \left(\frac{H}{2}\right) = \pi (4r^2) \left(\frac{H}{2}\right) = 2\pi r^2 H \]

5. **Relate the Volume to Mass**:
- The mass of water is given by:
\[ m = \rho V \]
- For the first capillary tube:
\[ m = \rho \cdot \pi r^2 H \]
- For the second capillary tube:
\[ m_2 = \rho \cdot 2\pi r^2 H \]

6. **Express the Mass in Terms of \( m \)**:
- We can express \( m_2 \) in terms of \( m \):
\[ m_2 = 2 \cdot (\rho \cdot \pi r^2 H) = 2m \]

### Conclusion:
The mass of water that will rise in the capillary tube of radius \( 2r \) is \( 2m \).

### Final Answer:
The mass of water that will rise in the capillary tube of radius \( 2r \) is **2m**.

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