When a long capillary tube is immersed in water, mass M of water rises in capillary. What mass of water will rise in another long capillary of half the radius and made of same material?

Options

A. `M`
B. `M/2`
C. `M/3`
D. `M/4`

Explanation

To solve the problem of how much mass of water will rise in a capillary tube of half the radius compared to a tube of radius \( r \), we can follow these steps:

### Step 1: Understand the relationship between mass of water and radius of the capillary
The mass of water that rises in a capillary tube is directly proportional to the radius of the tube. This is due to the balance of forces acting on the liquid column, where the upward force due to surface tension must balance the weight of the liquid.

### Step 2: Write the equation for the mass of water in terms of radius
Let \( M \) be the mass of water that rises in a capillary of radius \( r \). The relationship can be expressed as:
\[ M \propto r \]
This means that if we change the radius, the mass will change proportionally.

### Step 3: Calculate the mass for the new radius
If we have a new capillary tube with half the radius, \( r' = \frac{r}{2} \), we can express the new mass \( M' \) as:
\[ M' \propto r' = \frac{r}{2} \]
Thus, the new mass \( M' \) can be expressed in terms of \( M \):
\[ M' = \frac{M}{2} \]

### Step 4: Conclusion
Therefore, the mass of water that will rise in the capillary tube of half the radius is:
\[ M' = \frac{M}{2} \]

### Final Answer:
The mass of water that will rise in the capillary tube of half the radius is \( \frac{M}{2} \).

---