The vertical arms of U - tube have unequal raidus R and r `( R gt r)` . If a liquid of surface tension T and density `rho` rests in equillibrium inside the U - tube , find the level difference h between the meniscuses in the two arms .

Explanation

To find the level difference \( h \) between the meniscuses in the two arms of a U-tube with unequal radii \( R \) and \( r \) (where \( R > r \)), we will use the principles of fluid mechanics, particularly the effects of surface tension and hydrostatic pressure.

### Step-by-Step Solution:

1. **Understanding the Pressure at the Meniscus:**
The pressure at the meniscus in each arm of the U-tube is affected by the surface tension of the liquid. The pressure at the meniscus in the arm with radius \( R \) can be expressed as:
\[
P_1 = P_{\text{atm}} - \frac{2T}{R}
...