Notes : Define Torricelli's Law and Speed of Efflux | Derivation, Formula JEE, NEET

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Torricelli's Law and Speed of Efflux Class 11 Physics chapter 9 Mechanical Properties of Fluids Ncert Notes  | JEE, NEET & Board Exam Preparation - Physicskund 

Torricelli's Law is one of the most important applications of Bernoulli's theorem in fluid mechanics. It gives the speed with which a liquid flows out through a small hole made in a container. The law states that the speed of efflux of a liquid through a small orifice is equal to the speed acquired by a body falling freely through a height equal to the depth of the orifice below the liquid surface.

Introduction

When a small hole is made in the side of a vessel containing a liquid, the liquid comes out in the form of a jet. The speed with which the liquid emerges from the hole is called the speed of efflux.

The relationship between the speed of efflux and the depth of the hole below the liquid surface was first established by Evangelista Torricelli. This relationship is known as Torricelli's Law.

Speed of Efflux

The speed with which a liquid emerges from a small opening in a vessel is called the speed of efflux.

It is usually represented by: $v$

SI Unit:

$m\,s^{-1}$

Dimensional Formula:

$[M^0L^1T^{-1}]$

Arrangement of the Experiment

Consider a large tank filled with a liquid of density $\rho$.

Let:

  • Area of hole = $A_1$
  • Area of liquid surface = $A_2$
  • Velocity of liquid at hole = $v_1$
  • Velocity of liquid surface = $v_2$
  • Pressure at hole = $P_a$ (Atmospheric pressure)
  • Pressure at liquid surface = $P$
  • Height of hole from bottom = $y_1$
  • Height of liquid surface from bottom = $y_2$

If the hole is at a depth $h$ below the liquid surface, then

$$h=y_2-y_1$$

Equation of Continuity

The equation of continuity is based on the principle of conservation of mass.

For an incompressible fluid,

$$A_1v_1=A_2v_2$$

Therefore,

$$v_2=\frac{A_1}{A_2}v_1$$

Since the area of the tank is much larger than the area of the hole,

$$A_2\gg A_1$$

Hence,

$$\frac{A_1}{A_2}\approx0$$

Therefore,

$$v_2\approx0$$

Thus, the liquid surface falls very slowly and can be considered practically at rest.

Application of Bernoulli's Theorem

Apply Bernoulli's theorem between:

  • Point 1 → Hole
  • Point 2 → Liquid Surface

According to Bernoulli's theorem,

$$ P+\frac{1}{2}\rho v_2^2+\rho gy_2 = P_a+\frac{1}{2}\rho v_1^2+\rho gy_1 $$

Since

$$v_2\approx0$$

the kinetic energy term at the liquid surface becomes negligible.

Therefore,

$$ P+\rho gy_2 = P_a+\frac{1}{2}\rho v_1^2+\rho gy_1 $$

Derivation of the General Expression for Speed of Efflux

Rearranging the Bernoulli equation,

$$ \frac{1}{2}\rho v_1^2 = (P-P_a)+\rho g(y_2-y_1) $$

Since

$$ h=y_2-y_1 $$

therefore,

$$ \frac{1}{2}\rho v_1^2 = (P-P_a)+\rho gh $$

Multiplying both sides by 2,

$$ \rho v_1^2 = 2(P-P_a)+2\rho gh $$

Dividing both sides by $\rho$,

$$ v_1^2 = \frac{2(P-P_a)}{\rho} + 2gh $$

Taking square root,

$$ v_1= \sqrt{ \frac{2(P-P_a)}{\rho} + 2gh } $$

This is the general expression for the speed of efflux.

Special Case 1: Open Tank

For an open tank,

$$ P=P_a $$

Therefore,

$$ P-P_a=0 $$

Substituting into the general equation,

$$ v= \sqrt{2gh} $$

This equation is known as Torricelli's Law.

Statement of Torricelli's Law

The speed of efflux of a liquid through a small orifice is equal to the speed acquired by a body falling freely from rest through a vertical height equal to the depth of the orifice below the liquid surface.

Mathematical Form of Torricelli's Law

$$ v=\sqrt{2gh} $$

Where:

  • $v$ = Speed of efflux
  • $g$ = Acceleration due to gravity
  • $h$ = Depth of hole below liquid surface

Physical Significance of Torricelli's Law

For a freely falling body,

$$ v^2=u^2+2gh $$

For a body released from rest,

$$ u=0 $$

Therefore,

$$ v=\sqrt{2gh} $$

This is exactly the same expression obtained for the speed of efflux.

Hence, Torricelli's law shows that the speed of liquid emerging from a hole is equal to the speed acquired by a freely falling body through the same vertical height.

Key Points

  • Torricelli's law is derived using Bernoulli's theorem.
  • It applies to ideal fluids.
  • The hole should be very small compared to the tank area.
  • The speed of efflux increases with depth.
  • The speed of efflux is independent of the density of liquid in an open tank.
  • The law demonstrates conversion of potential energy into kinetic energy.
  • The velocity of the liquid surface is generally negligible.

Important Formulae

Depth of hole:

$$h=y_2-y_1$$

Continuity Equation:

$$A_1v_1=A_2v_2$$

General Speed of Efflux:

$$ v= \sqrt{ \frac{2(P-P_a)}{\rho} + 2gh } $$

Torricelli's Law:

$$ v=\sqrt{2gh} $$

Dimension of Speed:

$$ [M^0L^1T^{-1}] $$

Frequently Asked Questions (FAQ)

Q1. What is meant by speed of efflux?

The speed with which a liquid emerges from a small hole in a vessel is called the speed of efflux.

Q2. State Torricelli's law.

Torricelli's law states that the speed of efflux of a liquid through a small orifice is equal to the speed acquired by a body freely falling through a height equal to the depth of the orifice below the liquid surface.

Q3. What is the mathematical form of Torricelli's law?

$$v=\sqrt{2gh}$$

Q4. On which theorem is Torricelli's law based?

Torricelli's law is derived from Bernoulli's theorem.

Q5. Why is the velocity of the liquid surface neglected?

Because the area of the tank is much larger than the area of the hole ($A_2 \gg A_1$), the liquid surface moves very slowly. Therefore, $v_2 \approx 0$.

Q6. Does the speed of efflux depend on the density of liquid?

For an open tank, the speed of efflux is independent of the density of the liquid.

Q7. What is the SI unit of speed of efflux?

$$m\,s^{-1}$$

Q8. What is the dimensional formula of speed of efflux?

$$[M^0L^1T^{-1}]$$

Multiple Choice Questions (MCQs)

1. The speed of efflux from an open tank is

A) $\sqrt{gh}$
B) $\sqrt{2gh}$
C) $gh$
D) $2gh$

Answer: B) $\sqrt{2gh}$

2. Torricelli's law is derived using

A) Newton's Laws
B) Hooke's Law
C) Bernoulli's Theorem
D) Ohm's Law

Answer: C) Bernoulli's Theorem

3. The speed of efflux depends on

A) Shape of vessel
B) Colour of liquid
C) Depth below liquid surface
D) Atmospheric temperature only

Answer: C) Depth below liquid surface

4. The continuity equation is

A) $PV=constant$
B) $A_1v_1=A_2v_2$
C) $F=ma$
D) $P=\rho gh$

Answer: B) $A_1v_1=A_2v_2$

5. If the depth becomes four times, the speed of efflux becomes

A) Four times
B) One-fourth
C) Double
D) Half

Answer: C) Double

6. The SI unit of speed of efflux is

A) m
B) m/s
C) m/s²
D) kg/s

Answer: B) m/s

7. For an open tank, pressure at the liquid surface is

A) Zero
B) Infinite
C) Atmospheric Pressure
D) Vacuum Pressure

Answer: C) Atmospheric Pressure

8. Torricelli's law was proposed by

A) Newton
B) Archimedes
C) Pascal
D) Torricelli

Answer: D) Torricelli

Fill in the Blanks

1. The speed with which a liquid comes out of a hole is called __________.

Answer: Speed of Efflux

2. Torricelli's law is based on __________ theorem.

Answer: Bernoulli's

3. For an open tank, speed of efflux is __________.

Answer: $\sqrt{2gh}$

4. The SI unit of speed of efflux is __________.

Answer: m/s

5. The continuity equation is __________.

Answer: $A_1v_1=A_2v_2$

6. The speed of efflux is proportional to the square root of __________.

Answer: Depth (h)

7. When $A_2 \gg A_1$, velocity of the liquid surface is approximately __________.

Answer: Zero

8. The speed of efflux is independent of __________ of the liquid.

Answer: Density

True or False

1. Torricelli's law is derived from Bernoulli's theorem.

Answer: True

2. The speed of efflux depends on the density of liquid in an open tank.

Answer: False

3. The speed of efflux is equal to the speed acquired by a freely falling body through height h.

Answer: True

4. For an open tank, pressure at the liquid surface is atmospheric pressure.

Answer: True

5. The velocity of the liquid surface is generally greater than the velocity of efflux.

Answer: False

6. Continuity equation represents conservation of mass.

Answer: True

7. Speed of efflux increases with depth.

Answer: True

8. Torricelli's law applies only to gases.

Answer: False

Very Short Answer Questions (VSAQ)

Q1. What is speed of efflux?

Answer: The speed with which a liquid emerges from a small hole in a vessel is called the speed of efflux.

Q2. Who proposed Torricelli's law?

Answer: Evangelista Torricelli.

Q3. Write the formula for Torricelli's law.

Answer:

$v=\sqrt{2gh}$

Q4. On which theorem is Torricelli's law based?

Answer: Bernoulli's theorem.

Q5. What does h represent in Torricelli's law?

Answer: h represents the depth of the hole below the liquid surface.

Q6. What is the SI unit of speed of efflux?

Answer: $m\,s^{-1}$

Q7. What is the dimensional formula of speed of efflux?

Answer: $ [M^0L^1T^{-1}]$

Q8. What is the pressure at the liquid surface of an open tank?

Answer: Atmospheric pressure.

Q9. What is the continuity equation?

Answer:

$A_1v_1=A_2v_2$

Q10. What happens to the speed of efflux if depth increases?

Answer: The speed of efflux increases.

Short Answer Questions (SAQ)

Q1. State Torricelli's law.

Answer:

Torricelli's law states that the speed of efflux of a liquid through a small orifice is equal to the speed acquired by a body falling freely from rest through a vertical height equal to the depth of the orifice below the liquid surface.

Mathematically,

$v=\sqrt{2gh}$

Q2. Why is the velocity of the liquid surface neglected during derivation?

Answer:

According to the continuity equation,

$A_1v_1=A_2v_2$

Since the area of the liquid surface is much larger than the area of the hole,

$A_2\gg A_1$

Therefore,

$v_2\approx0$

Hence the velocity of the liquid surface is negligible.

Q3. What is the physical significance of Torricelli's law?

Answer:

Torricelli's law shows that the speed of a liquid emerging from a hole is equal to the speed acquired by a body freely falling through the same vertical height.

It demonstrates the conversion of gravitational potential energy into kinetic energy.

Q4. Write the continuity equation and explain its significance.

Answer:

The continuity equation is

$A_1v_1=A_2v_2$

It is based on the principle of conservation of mass. The mass of fluid entering a region per second is equal to the mass leaving that region per second.

Q5. Why does the speed of efflux increase with depth?

Answer:

As the depth increases, hydrostatic pressure increases.

Greater pressure provides greater kinetic energy to the liquid, thereby increasing the speed of efflux.

Since

$v=\sqrt{2gh}$

the speed is directly proportional to the square root of depth.

Long Answer Questions (LAQ)

Q1. Derive Torricelli's law using Bernoulli's theorem.

Answer:

Consider a large tank containing a liquid. Let:

  • Area of hole = $A_1$
  • Area of liquid surface = $A_2$
  • Velocity at hole = $v_1$
  • Velocity at surface = $v_2$
  • Depth of hole below surface = $h$

From the continuity equation,

$A_1v_1=A_2v_2$

Since

$A_2\gg A_1$

therefore,

$v_2\approx0$

Applying Bernoulli's theorem between the liquid surface and the hole,

$$ P+\frac{1}{2}\rho v_2^2+\rho gy_2 = P_a+\frac{1}{2}\rho v_1^2+\rho gy_1 $$

Since

$$v_2\approx0$$

and for an open tank,

$$P=P_a$$

Therefore,

$$ \rho gy_2 = \frac{1}{2}\rho v_1^2+\rho gy_1 $$

Rearranging,

$$ \frac{1}{2}\rho v_1^2 = \rho g(y_2-y_1) $$

Since

$$ h=y_2-y_1 $$

Hence,

$$ \frac{1}{2}\rho v_1^2 = \rho gh $$

Therefore,

$$ v_1=\sqrt{2gh} $$

This equation is called Torricelli's Law.

Q2. Explain the physical meaning of Torricelli's law.

Answer:

According to Torricelli's law,

$$v=\sqrt{2gh}$$

For a freely falling body,

$$v^2=u^2+2gh$$

If the body starts from rest,

$$u=0$$

Therefore,

$$v=\sqrt{2gh}$$

Thus, the speed of efflux of a liquid is equal to the speed acquired by a body falling freely through the same height.

The law represents the conversion of gravitational potential energy into kinetic energy.

Q3. Derive the general expression for speed of efflux for a pressurized tank.

Answer:

Applying Bernoulli's theorem between the liquid surface and the hole,

$$ P+\rho gy_2 = P_a+\frac{1}{2}\rho v^2+\rho gy_1 $$

Rearranging,

$$ \frac{1}{2}\rho v^2 = (P-P_a)+\rho g(y_2-y_1) $$

Since

$$ h=y_2-y_1 $$

Therefore,

$$ \frac{1}{2}\rho v^2 = (P-P_a)+\rho gh $$

Multiplying by 2,

$$ \rho v^2 = 2(P-P_a)+2\rho gh $$

Dividing by $\rho$,

$$ v^2 = \frac{2(P-P_a)}{\rho} + 2gh $$

Hence,

$$ v= \sqrt{ \frac{2(P-P_a)}{\rho} + 2gh } $$

This is the general expression for the speed of efflux from a pressurized tank.

Chapter Summary

  • Speed of efflux is the speed with which a liquid emerges from a small hole.
  • Torricelli's law is derived using Bernoulli's theorem.
  • For an open tank:

$$v=\sqrt{2gh}$$

  • The speed of efflux is equal to the speed acquired by a freely falling body through height h.
  • The speed increases with depth.
  • The speed is independent of the density of the liquid.
  • The continuity equation is:

$$A_1v_1=A_2v_2$$

  • For a pressurized tank:

$$ v= \sqrt{ \frac{2(P-P_a)}{\rho} + 2gh } $$

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