Question

A small hole of radius 0.1 mm is present at the bottom of a tumbler. Up to what maximum height may water be stored inside the tumbler so that no water leaks out through the whole? [ given that the surface tension of water is 7.1 x 10^(-2) N/m ]

Answer 1

A small hole of radius $0.1 \, \text{mm}$ is present at the bottom of a tumbler. Up to what maximum height may water be stored inside the tumbler so that no water leaks out through the whole? [given that the surface tension of water is $7.1 \times 10^4$(-2) N/m]

Solution

We are given:

$$r = 0.1 \, \text{mm} = 10^{-4} \, \text{m}$$ $$T = 7.1 \times 10^{-2} \, \text{N/m}$$

Force of surface tension can be calculated as:

$$F_{\text{tension}} = 2\pi r \times T$$

Force on water surface due to weight of the water in a tumbler is:

$$F_{\text{weight}} = p \times \pi r^2$$ $$p = \rho \times g \times h$$

For maximum height:

$$F_{\text{weight}} = F_{\text{tension}}$$ $$2\pi r T = \rho g h_{\max} \pi r^2$$

Thus:

$$h_{\max} = \frac{2T}{\rho g r}$$

Calculation:

$$h_{\max} = \frac{2T}{\rho g r} = \frac{2 \times 7.1 \times 10^{-2}}{1000 \times 9.8 \times 10^{-4}} \approx 0.145 \, \text{m} = 14.5 \, \text{cm}$$

Answer: 0.145 m