Question

Water is filled in a tank up to a height of 20 cm from the bottom of the tank. Water flows through a hole of area $1\text{ mm}^2$ at its bottom. The mass of the water coming out from the hole in a time of 0.6 s is
(Density of water $= 1000\text{ kg m}^{-3}$ and acceleration due to gravity $= 10\text{ ms}^{-2}$)

• A. $1.8\text{ g}$
• B. $1.2\text{ g}$
• C. $0.6\text{ g}$
• D. $2.4\text{ g}$

Hint:

Remember standard conversions: $1\text{ mm}^2 = 10^{-6}\text{ m}^2$ and $1\text{ kg} = 1000\text{ g}$.

Answer 1

The Correct Option is B

Step 1: Calculate Velocity of Efflux:
Using Torricelli's Law: $v = \sqrt{2gh}$. $h = 20\text{ cm} = 0.2\text{ m}$. $g = 10\text{ m/s}^2$. \[ v = \sqrt{2 \times 10 \times 0.2} = \sqrt{4} = 2\text{ m/s} \]
Step 2: Calculate Volume Flow Rate:
$Q = \text{Area} \times \text{Velocity}$. Area $A = 1\text{ mm}^2 = 1 \times 10^{-6}\text{ m}^2$. \[ Q = 10^{-6} \times 2 = 2 \times 10^{-6}\text{ m}^3/\text{s} \]
Step 3: Calculate Mass Flow Rate:
$\dot{m} = \text{Density} \times Q$. $\rho = 1000\text{ kg/m}^3$. \[ \dot{m} = 1000 \times 2 \times 10^{-6} = 2 \times 10^{-3}\text{ kg/s} = 2\text{ g/s} \]
Step 4: Calculate Total Mass:
Mass $m = \dot{m} \times t$. Given $t = 0.6\text{ s}$. \[ m = 2\text{ g/s} \times 0.6\text{ s} = 1.2\text{ g} \]