A glass tube of 1 mm bore is dipped vertically into a container of mercury, with its lower end 5 cm below the mercury surface. What must be the gauge pressure of air in the tube to a hemispherical bubble at its lower end? Given density of mercury ` = 13.6 xx 10^(3) kg//m^(3)`, surface tension of mercury = `440 xx 10^(-3) Nm^(-1)` and `g = 10m//s^(2)`

Explanation

To solve the problem, we need to find the gauge pressure of air in the tube that will support a hemispherical bubble at its lower end. The gauge pressure can be calculated by considering two contributions: the hydrostatic pressure due to the height of mercury and the pressure contribution due to the surface tension of the mercury bubble.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Density of mercury, \( \rho = 13.6 \times 10^3 \, \text{kg/m}^3 \)
- Surface tension of mercury, \( S = 440 \times 10^{-3} \, \text{N/m} \)
- Acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \)
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