Consider a completely full cylindrical water tank of height 1.6 m and cross-sectional area 0.5 m^2. It has a small hole in its side at a height 90 cm from the bottom. Assume, the cross-sectional area…

Given: Height of tank = 1.6 m, area of tank = 0.5 m^2, hole is at 0.9 m from the bottom, load applied = 50 kg, and g = 10 m/s^2. Find: Velocity of water coming out of the hole. Use Bernoulli's equation or Torricelli's theorem with the extra pressure due to the applied load. Since the tank area is large, velocity of the upper surface is negligible. The pressure head due to the applied load is obtained from h extra = F load g A tank where F load = mg = 50 10 = 500 N So, h extra = 5001000 10 0.5 = 0.1 m The water column above the hole has height h = 1.6 - 0.9 = 0.7 m Therefore the total effective head is h total = 0.7 + 0.1 = 0.8 m Now apply Torricelli's theorem: v = 2gh total Substituting the values, v = 2 10 0.8 = 16 = 4 m/s Therefore, the velocity of water coming out is 4 m/s and the correct option is D. A compact Bernoulli form shown in the solution is: P1 + 12 v1^2 + gh = P2 + 12 v2^2 With v1 0, P1 = P0 + mgA, and P2 = P0, we get the same result.

Consider a completely full cylindrical water tank of height 1.6 m… — Solution | JEE Physics

MCQMediumJEE 2025Bernoulli's Theorem

Consider a completely full cylindrical water tank of height $1.6 \, \text{m}$ and cross-sectional area $0.5 \, \text{m}^2$ . It has a small hole in its side at a height $90 \, \text{cm}$ from the bottom. Assume, the cross-sectional area of the hole to be negligibly small as compared to that of the water tank. If a load $50 \, \text{kg}$ is applied at the top surface of the water in the tank then the velocity of the water coming out at the instant when the hole is opened is : ( $g = 10 \, \text{m/s}^2$ )

A
$3 \, \text{m/s}$
B
$5 \, \text{m/s}$
C
$2 \, \text{m/s}$
D
$4 \, \text{m/s}$

Answer

Correct answer:D

Step-by-step solution

Standard Method

Given: Height of tank = $1.6 \, \text{m}$ , area of tank = $0.5 \, \text{m}^2$ , hole is at $0.9 \, \text{m}$ from the bottom, load applied = $50 \, \text{kg}$ , and $g = 10 \, \text{m/s}^2$ .

Find: Velocity of water coming out of the hole.

Use Bernoulli's equation or Torricelli's theorem with the extra pressure due to the applied load. Since the tank area is large, velocity of the upper surface is negligible.

The pressure head due to the applied load is obtained from

h_{\text{extra}} = \frac{F_{\text{load}}}{\rho g A_{\text{tank}}}

where

F_{\text{load}} = mg = 50 \times 10 = 500 \, \text{N}

So,

h_{\text{extra}} = \frac{500}{1000 \times 10 \times 0.5} = 0.1 \, \text{m}

Bernoulli Equation Expansion

The water column above the hole has height

h = 1.6 - 0.9 = 0.7 \, \text{m}

Therefore the total effective head is

h_{\text{total}} = 0.7 + 0.1 = 0.8 \, \text{m}

Effective Head Trick

Now apply Torricelli's theorem:

v = \sqrt{2gh_{\text{total}}}

Substituting the values,

v = \sqrt{2 \times 10 \times 0.8} = \sqrt{16} = 4 \, \text{m/s}

Therefore, the velocity of water coming out is $4 \, \text{m/s}$ and the correct option is D.

A compact Bernoulli form shown in the solution is:

P_1 + \frac{1}{2}\rho v_1^2 + \rho gh = P_2 + \frac{1}{2}\rho v_2^2

With $v_1 \approx 0$ , $P_1 = P_0 + \frac{mg}{A}$ , and $P_2 = P_0$ , we get the same result.

Common mistakes

Ignoring the pressure due to the applied $50 \, \text{kg}$ load. This is wrong because the load adds an extra pressure head on the water surface. Convert it into equivalent water head using $h_{\text{extra}} = \frac{F}{\rho g A}$ .
Using the full tank height $1.6 \, \text{m}$ as the head. This is wrong because only the height difference between the top surface and the hole contributes. Use $1.6 - 0.9 = 0.7 \, \text{m}$ before adding the extra pressure head.
Taking the velocity at the top surface equal to the velocity at the hole. This is wrong because the tank cross-sectional area is much larger than the hole area, so the surface speed is negligible compared to the efflux speed.

Practice more Bernoulli's Theorem questions

Get unlimited AI-adaptive practice, mastery tracking, and an AI tutor that explains every step — free to start.

Start free trial More on Bernoulli's Theorem

Answer

Step-by-step solution

Common mistakes

Practice more Bernoulli's Theorem questions

Related questions