Question

A large tank open to atmosphere at top and filled with water, develops a small hole in the side at a point \(20 \, \text{m}\) below the water level. If the rate of flow of water from the hole is \(3\times 10^{-3}\,\text{m}^3/\text{min}\), then the area of hole is \((g=10\,\text{m s}^{-2})\):

• A. \(4\,\text{mm}^2\)
• B. \(1.5\,\text{mm}^2\)
• C. \(2.5\,\text{mm}^2\)
• D. \(2\,\text{mm}^2\)

Hint:

For water flowing out of a small hole in a tank, first use Torricelli's theorem \(v=\sqrt{2gh}\), then use \(Q=Av\) to find the area of the hole.

Answer 1

The Correct Option is C

Step 1: Find the efflux velocity using Torricelli's theorem.
\[ v = \sqrt{2gh} = \sqrt{2\times10\times20} = 20\,\text{m s}^{-1} \]

Step 2: Find area from the flow rate.
Flow rate \( Q = \frac{3\times10^{-3}}{60} = 5\times10^{-5}\,\text{m}^3\text{s}^{-1} \). \[ A = \frac{Q}{v} = \frac{5\times10^{-5}}{20} = 2.5\times10^{-6}\,\text{m}^2 = 2.5\,\text{mm}^2 \] \[ \boxed{2.5\,\text{mm}^2} \]