The radii of the two columns is U-tube are `r_(1)` and `r_(2)(r_(2)gtr_(1))`. When a liquid of density `rho` (angle of contact is `0^@))` is filled in it, the level different of liquid in two arms is h. The surface tension of liquid is
`(g=` acceleration due to gravity)

Options

A. `(rho gh r_(1)r_(2))/(2(r_(2) - r_(1)))`
B. `(rho gh (r_(2) - r_(1)))/(2r_(1) r_(2))`
C. `(2 (r_(2) - r_(1)))/(rho gh r_(1) r_(2))`
D. `(rho gh)/(2(r_(2) - r_(1)))`

Answer

To find the surface tension of the liquid in the U-tube, we can follow these steps:

### Step 1: Understand the Problem
We have a U-tube with two arms of different radii, \( r_1 \) and \( r_2 \) (where \( r_2 > r_1 \)). A liquid of density \( \rho \) is filled in the U-tube, and the angle of contact is \( 0^\circ \). The height difference between the two arms is \( h \).

### Step 2: Use the Formula for Height due to Surface Tension
The height to which the liquid rises in each arm due to surface tension can be expressed as:
\[ h_1 = \frac{2T \cos \theta}{\rho g r_1} \]
\[ h_2 = \frac{2T \cos \theta}{\rho g r_2} \]
Since the angle of contact \( \theta = 0^\circ \), we have \( \cos 0^\circ = 1 \). Therefore, the equations simplify to:
\[ h_1 = \frac{2T}{\rho g r_1} \]
\[ h_2 = \frac{2T}{\rho g r_2} \]

### Step 3: Relate the Heights to the Given Height Difference
The total height difference \( h \) between the two arms can be expressed as:
\[ h = h_1 - h_2 \]
Substituting the expressions for \( h_1 \) and \( h_2 \):
\[ h = \frac{2T}{\rho g r_1} - \frac{2T}{\rho g r_2} \]

### Step 4: Factor Out Common Terms
Factoring out the common terms from the right side:
\[ h = \frac{2T}{\rho g} \left( \frac{1}{r_1} - \frac{1}{r_2} \right) \]

### Step 5: Solve for Surface Tension \( T \)
Rearranging the equation to solve for \( T \):
\[ T = \frac{h \rho g}{2} \left( \frac{1}{\frac{1}{r_1} - \frac{1}{r_2}} \right) \]
This can be further simplified:
\[ T = \frac{h \rho g r_1 r_2}{2(r_2 - r_1)} \]

### Final Answer
Thus, the surface tension \( T \) of the liquid is given by:
\[ T = \frac{\rho g h r_1 r_2}{2(r_2 - r_1)} \]