If 'M' is the mass of water that rises in a capillary tube of radius 'r', then mass of water which will ris in a capillary tube of radius '2r' is :

Options

A. M
B. `(M)/(2)`
C. `4 M`
D. `2 M`

Answer

To solve the problem, we need to analyze the relationship between the mass of water that rises in a capillary tube and the radius of the tube. Here's a step-by-step solution:

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

From this formula, we can see that the height \( h \) is inversely proportional to the radius \( r \):
\[ h \propto \frac{1}{r} \]

### Step 2: Relate the heights for different radii
Let’s denote:
- \( r_1 = r \) (the radius of the first tube),
- \( h_1 = h \) (the height of water in the first tube),
- \( r_2 = 2r \) (the radius of the second tube).

Using the inverse relationship:
\[ \frac{h_1}{h_2} = \frac{r_2}{r_1} = \frac{2r}{r} = 2 \]
This implies:
\[ h_2 = \frac{h_1}{2} = \frac{h}{2} \]

### Step 3: Calculate the mass of water in the capillary tubes
The mass of water \( M \) that rises in a capillary tube can be expressed as:
\[ M = \text{Volume} \times \text{Density} \]
The volume \( V \) of the water column in a cylindrical tube is given by:
\[ V = \pi r^2 h \]
Thus, for the first tube:
\[ M_1 = \pi r^2 h \cdot \rho \]
For the second tube with radius \( 2r \) and height \( h_2 = \frac{h}{2} \):
\[ M_2 = \pi (2r)^2 \left(\frac{h}{2}\right) \cdot \rho \]
Calculating \( M_2 \):
\[ M_2 = \pi (4r^2) \left(\frac{h}{2}\right) \cdot \rho = 2 \cdot \pi r^2 h \cdot \rho = 2M \]

### Conclusion
Thus, the mass of water that rises in a capillary tube of radius \( 2r \) is:
\[ M_2 = 2M \]

### Final Answer
The mass of water that will rise in a capillary tube of radius \( 2r \) is \( 2M \).