A capillary tube of length (i) `l = 60 cm`, (ii) `l = 50 cm` and radius `r = 1//4 mm` is immersed vertically into water. Find the capillary rise in both cases. Angle of contact `= 0^@`. Take coefficient of surface tension as `72 "dyne"//cm, g = 1000 cm s^(-12)`.

Explanation

To solve the problem of finding the capillary rise in two different lengths of a capillary tube, we will use the formula for capillary rise:

\[ h = \frac{2T \cos \theta}{\rho r g} \]

Where:
- \( h \) = capillary rise
- \( T \) = surface tension (given as 72 dyne/cm)
- \( \theta \) = angle of contact (given as \( 0^\circ \))
- \( \rho \) = density of water (approximately \( 1 \, \text{g/cm}^3 \) or \( 1000 \, \text{kg/m}^3 \))
- \( r \) = radius of the capillary tube (given as \( \frac{1}{4} \, \text{mm} = 0.25 \, \text{mm} = 0.025 \, \text{cm} \))
- \( g \) = acceleration due to gravity (given as \( 1000 \, \text{cm/s}^2 \))

### Step 1: Convert the radius to centimeters
The radius \( r \) is given as \( \frac{1}{4} \, \text{mm} \):
\[ r = \frac{1}{4} \, \text{mm} = 0.25 \, \text{mm} = 0.025 \, \text{cm} \]

### Step 2: Substitute the values into the capillary rise formula
Using the values:
- \( T = 72 \, \text{dyne/cm} \)
- \( \theta = 0^\circ \) (thus, \( \cos 0 = 1 \))
- \( \rho = 1 \, \text{g/cm}^3 \)
- \( r = 0.025 \, \text{cm} \)
- \( g = 1000 \, \text{cm/s}^2 \)

Substituting these into the formula:
\[ h = \frac{2 \times 72 \times \cos(0)}{1 \times 0.025 \times 1000} \]
\[ h = \frac{2 \times 72 \times 1}{0.025 \times 1000} \]
\[ h = \frac{144}{25} = 5.76 \, \text{cm} \]

### Step 3: Calculate the capillary rise for the first case (l = 60 cm)
Since the calculated capillary rise \( h = 5.76 \, \text{cm} \) is less than the length of the capillary tube \( l = 60 \, \text{cm} \), the capillary rise in this case will be:
\[ h_1 = 5.76 \, \text{cm} \]

### Step 4: Calculate the capillary rise for the second case (l = 50 cm)
In the second case, the calculated capillary rise \( h = 5.76 \, \text{cm} \) is also less than the length of the capillary tube \( l = 50 \, \text{cm} \). Therefore, the capillary rise in this case will be:
\[ h_2 = 5.76 \, \text{cm} \]

### Final Results
- For the first case (length = 60 cm), the capillary rise is \( 5.76 \, \text{cm} \).
- For the second case (length = 50 cm), the capillary rise is also \( 5.76 \, \text{cm} \).