NotesShala Class 11 · Physics

Chapter13

Oscillations

Periodic Motion and SHM · Characteristics of SHM · Energy in SHM, Spring and Pendulum

Reading key Definition Note Remember Derivation Strategy Formulas sit on graph paper · coloured words are the key terms

Part 1

Periodic Motion and SHM

An object is in motion when its position changes with time, as seen from a reference point. We can sort motions into types by the path they follow and by some other properties.

1.1Periodic and Oscillatory Motion

Three types of motion

iNon-repetitive

The motion does not repeat.

e.g. Rectilinear motion, projectile motion.

iiPeriodic

The motion repeats after equal times.

e.g. Circular or orbital motion, a bouncing ball.

iiiOscillatory

The body moves to and fro about a point.

e.g. The motion of a pendulum.

Definition

Periodic Motion

Periodic motion is a motion that repeats itself after equal intervals of time.

e.g. The planets revolve around the Sun. The Earth rotates about its polar axis.

Definition

Oscillatory Motion

Oscillatory motion is the to-and-fro (back and forth) motion of a body about a fixed point, repeated at regular intervals of time.

This fixed point is called the mean position, the equilibrium position or the centre of oscillation.

e.g. A simple pendulum swings to and fro about its mean position.

Oscillations Around Us

The strings of a sitar or a guitar vibrate.
Air molecules vibrate and carry sound from place to place.
Atoms in a solid vibrate about their equilibrium positions.
An AC voltage keeps going positive and negative about its mean value.

A planet repeats its path, so its motion is periodic. A ball in a bowl moves to and fro about O, so its motion is oscillatory

Remember

Periodic vs Oscillatory Motion

Every oscillatory motion is periodic. But a periodic motion need not be oscillatory.
e.g. A planet moving around the Sun is in periodic motion. It is not oscillatory, because it does not move to and fro.

Note

Equilibrium Position

An oscillating body has an equilibrium position on its path. At this position, no net force acts on the body. So a body placed there at rest stays there for ever.

Definition

Restoring Force

The force that tries to bring a displaced body back to its equilibrium position is called the restoring force.

Give the body a small displacement from its mean position. The restoring force pulls it back. This happens again and again, and the body oscillates.

e.g. A ball rests at the bottom of a bowl in equilibrium. Push it a little to one side. It starts oscillating inside the bowl.

1.2Period and Frequency

Period

Definition

The period (time period) is the smallest time interval after which a periodic motion repeats itself. It is written as $T$ . Its SI unit is the second (s).

Periods can be very short or very long. So a suitable unit is chosen in each case:

(i) Period of vibration of a quartz crystal, in microseconds (

1\ \mu\text{s} = 10^{-6}\ \text{s}

)μs

(ii) The planet Mercury goes once round the Sun in88 Earth days

(iii) Halley's comet takes 76 years for one trip round the Sun, so we see itevery 76 years

Frequency

Definition

The frequency of an oscillating body is the number of oscillations it completes about its mean position in unit time. It is written as $\nu$ .

Its SI unit is the hertz (Hz). The unit is named after Heinrich Rudolph Hertz (1857–1894), who discovered radio waves.

Frequency is also the reciprocal of the time period:

Frequency

\nu = \dfrac{1}{T}

where, $T$ = time period

1\ \text{hertz} = 1\ \text{Hz} = 1\ \text{oscillation per second} = 1\ \text{s}^{-1}

Definition

Vibrational Motion

An oscillatory motion of very high frequency is called vibrational motion.

1.3Displacement

Displacement always means a change in position of a particle. Only the reference point is different in oscillations.

iIn General

Displacement is the change in position of a particle measured from the origin.

iiIn Oscillations

Displacement is the change in position measured from the mean position (equilibrium position). This point is also called the reference position.

1.4Displacement Variable

The quantity that measures the displacement during an oscillation is called the displacement variable. It changes with time. It takes both positive (+) and negative (−) values.

Two examples

iBlock on a Spring

A block is fixed to one end of a spring. The other end of the spring is fixed to a rigid wall. The displacement variable $x$ is the distance of the block from its mean position at any time. It is a linear deviation.

x

= linear deviation of the block from its mean position

iiSimple Pendulum

For a simple pendulum, the displacement variable is the angle $\theta$ that the string makes with the vertical at any time. It is an angular deviation.

\theta

= angular deviation of the pendulum from the vertical

1.5Periodic Function

Definition

We can write the displacement as a mathematical function of time. A function of time that repeats its value after a regular interval (its period) is called a periodic function.

The simplest periodic function is:

f(t) = A\cos\omega t \qquad \ldots\text{(i)}

Its period is $T = 2\pi/\omega$ . The reason is simple. Adding $2\pi$ to the angle $\omega t$ does not change the cosine. So the function takes the same value again after a time $T$ :

Periodic Function

f(t) = f(t+T), \qquad T = \dfrac{2\pi}{\omega}

In the same way, the sine function $f(t) = A\sin\omega t$ is periodic. A linear combination of sine and cosine functions is also periodic, with the same period $T$ :

f(t) = A\sin\omega t + B\cos\omega t \qquad \ldots\text{(ii)}

Now put:

Coefficient of

\sin\omega t

A = D\cos\phi

Coefficient of

\cos\omega t

B = D\sin\phi

Then Eq. (ii) takes a simpler form:

Combined Periodic Function

f(t) = D\sin(\omega t + \phi)

where, $D = \sqrt{A^2 + B^2}$ and $\phi = \tan^{-1}\!\left(\dfrac{B}{A}\right)$

Good to know

Fourier's Result

The French mathematician Jean Baptiste Joseph Fourier (1768–1830) gave an important result. Any periodic function can be written as a superposition of sine and cosine functions. These functions have different time periods and suitable coefficients.

1.6Simple Harmonic Motion

Definition

Simple harmonic motion (SHM) is a special type of periodic motion. In it, a particle moves to and fro about a mean position under a restoring force. This force is directly proportional to the displacement and always acts towards the mean position.

The Restoring Force in SHM

iDirection

It always acts towards the mean position.

iiMagnitude

At every instant, its size is directly proportional to how far the particle is from the mean position.

KeyBecause of such a restoring force, the displacement in SHM always changes with time as a sine or cosine, that is, as a sinusoidal function.

1.7Equation of SHM

A particle moves back and forth along the $x$ -axis about the origin O. It stays between the limits $+A$ and $-A$ .

This motion is SHM if the displacement $x$ of the particle from the origin changes with time as given below.

A particle in SHM between

-A

and

+A

Displacement in SHM

x(t) = A\cos(\omega t + \phi)

where, $A$ , $\omega$ and $\phi$ are constants

With

\phi = 0

: at

t = 0

, the particle is at

x = +A

t = T

, it comes back to the same point

x = +A

Let the particle be at point P at an instant $t$ . Take O as the mean position.

Then $OP = x$ tells how far the particle is from the mean position. It is the displacement of the particle.

Displacement

OP = x

from O

At that instant, a restoring force $F$ acts on the particle:

Force Law for SHM

F = -kx

where, $k$ = force constant. Its SI unit is $\text{N m}^{-1}$ (N/m).

Note

Why the Minus Sign?

The minus sign shows that $F$ always points towards the mean position. So $F$ and $x$ are always in opposite directions. The relation $F = -kx$ is called the force law for SHM.

1.8Position of the Particle at Different Times

The figure shows where a particle in SHM is at different times. It starts at $+A$ at $t = 0$ . Each step is a time interval of $\dfrac{T}{4}$ .

Positions at

t = 0,\ \tfrac{T}{4},\ \tfrac{T}{2},\ \tfrac{3T}{4},\ T,\ \tfrac{5T}{4}

. The speed is zero at

\pm A

and maximum at O

Observations

1Repetition

The motion repeats itself after a time $T$ .

2Fixed Period

$T$ stays fixed. It does not change if the starting position (at $t = 0$ ) is changed.

1.9SHM and Uniform Circular Motion

This gives a geometrical picture of SHM.

A particle P starts from point X. It moves with uniform speed along a circle of radius $A$ and centre O.

Draw a perpendicular from P to the diameter XOX′. Its foot is the point P′.

Circle of reference: P′ is the foot of the perpendicular from P on

XOX'

Circle of Reference

The circle on which P moves is called the circle of reference.

Reference Particle

The particle P is called the reference particle.

Projection

P′ is called the projection of P on the diameter XOX′.

As P goes round the circle, its projection P′ moves along the diameter as follows:

P moves along the circle	P′ moves along the diameter
$X \rightarrow Y$	$X \rightarrow O$
$Y \rightarrow X'$	$O \rightarrow X'$
$X' \rightarrow Y'$	$X' \rightarrow O$
$Y' \rightarrow X$	$O \rightarrow X$

Result

P goes round the circle. At the same time, P′ moves to and fro about O along XOX′. So the motion of P′ about O is simple harmonic.

Definition

Geometrical Definition of SHM

The projection of a uniform circular motion on any diameter of its circle is a simple harmonic motion.

Note

The projection of the reference particle on any other diameter of the circle of reference also moves in SHM.

Part 2

Characteristics of SHM

A few key quantities describe every simple harmonic motion: displacement, amplitude, phase, angular frequency, velocity and acceleration.

2.1Displacement in SHM

Definition

The displacement of a particle in SHM at any instant is its distance from the mean position at that instant.

The displacement changes continuously with time. So we can draw it as a graph.

The graph swings between $+A$ and $-A$ about O.

Displacement

x(t) = A\cos(\omega t + \phi)

Displacement as a continuous function of time

Definition

Sinusoidal Function

A function that contains a sine or a cosine term is called a sinusoidal function.

2.2Amplitude

Definition

The amplitude of a particle in SHM is the magnitude of its maximum displacement from the mean position. It is measured on either side of the mean position.

Two SHMs can have the same angular frequency $\omega$ and the same phase constant $\phi$ . But their amplitudes $A$ and $B$ can be different.

Same

\omega,\ \phi

Differentamplitudes

A,\ B

Same

\omega

and

\phi

, different amplitudes

A

and

B

sin or cos goes from +1 to −1⇒

x

goes from

+A

-A

2.3Phase

Definition

The phase of an SHM is the quantity $(\omega t + \phi)$ inside its sinusoidal function. For a fixed amplitude $A$ , the phase decides the state of motion, that is, the position and the velocity of the particle at time $t$ .

t = 0

⇒phase

= \phi

⇒

\phi

is the phase constant (phase angle)

Two SHMs can have the same amplitude $A$ and the same angular frequency $\omega$ . But their phase constants $\phi$ can be different.

In the graph, curve 1 has $\phi = \dfrac{\pi}{2}$ . Curve 2 has $\phi = -\dfrac{\pi}{4}$ . Both curves have amplitude $A$ .

Same

A

and

\omega

, different phase constants

2.4Angular Frequency

Definition

The angular frequency of a periodic motion is $2\pi$ times its frequency. It is written as $\omega$ . Its SI unit is the radian per second ( $\text{rad s}^{-1}$ ).

T = \dfrac{2\pi}{\omega} \;\Rightarrow\; \omega = \dfrac{2\pi}{T} = 2\pi\nu, \quad \text{where } \nu = \text{frequency of the particle}

Derivation

Why is

\omega = 2\pi/T

The motion repeats after one period $T$ . So the displacement must come back to its first value after $T$ , that is, $x(t) = x(t+T)$ for every $t$ . Take $x(t) = A\cos\omega t$ :

$A\cos\omega t = A\cos\omega(t+T)$
The cosine first repeats when its angle grows by $2\pi$ . So:
$\omega(t+T) = \omega t + 2\pi$
$\omega T = 2\pi \;\Rightarrow\; \omega = \dfrac{2\pi}{T}$

Angular Frequency

\omega = \dfrac{2\pi}{T} = 2\pi\nu

$\omega$ is $2\pi$ times the frequency of oscillation, $\nu = \dfrac{1}{T}$

Two SHMs can have the same amplitude $A$ and the same phase constant $\phi$ . But their angular frequencies $\omega$ can be different.

In the graph, curve (b) has half the period of curve (a). So it has twice the frequency of curve (a).

Same

A

and

\phi

: curve (b) has half the period of curve (a)

2.5Velocity in SHM

Definition

The velocity of a particle in SHM at any instant is the rate of change of its displacement with time at that instant.

Velocity in Terms of Displacement

v = \omega\sqrt{A^2 - x^2}

This gives the magnitude of the velocity (the speed) at displacement $x$ .

At the Mean Position (

x = 0

)

v = \omega A

The velocity is maximum.

At an Extreme Position (

x = \pm A

)

v = 0

The velocity is minimum (zero).

We can also find the velocity by differentiating $x(t)$ :

v(t) = \dfrac{d}{dt}[x(t)] = \dfrac{d}{dt}[A\cos(\omega t + \phi)]

Velocity as a Function of Time

v(t) = -\omega A\sin(\omega t + \phi)

2.6Acceleration in SHM

Definition

The acceleration of a particle in SHM at any instant is the rate of change of its velocity with time at that instant.

Acceleration in Terms of Displacement

a = -\omega^2 x

At the Mean Position (

x = 0

)

a = 0

The acceleration is minimum (zero).

At an Extreme Position (

x = \pm A

)

a = \mp\,\omega^2 A

The acceleration is maximum in size.

Remember

Velocity and Acceleration Behave Oppositely

At the mean position: velocity is maximum, acceleration is zero.
At an extreme position: velocity is zero, acceleration is maximum.

We can also find the acceleration by differentiating $v(t)$ :

a(t) = \dfrac{d}{dt}[v(t)] = \dfrac{d}{dt}[-\omega A\sin(\omega t + \phi)]

Acceleration as a Function of Time

a(t) = -\omega^2 A\cos(\omega t + \phi)

2.7Graphs of Displacement, Velocity and Acceleration

Take $\phi = 0$ . Then the three quantities of a body in SHM are:

Displacement

x(t) = A\cos\omega t

Velocity

v(t) = -\omega A\sin\omega t

Acceleration

a(t) = -\omega^2 A\cos\omega t

The three graphs share one time axis.

v

reaches each value

\tfrac{T}{4}

earlier than

x

(it leads by

\tfrac{\pi}{2}

), and

a

is always opposite to

x

Quantity	Range of values	Phase difference from $x(t)$
Displacement $x(t)$	$-A$ to $+A$	Zero (reference curve)
Velocity $v(t)$	$-\omega A$ to $+\omega A$	$\dfrac{\pi}{2}$ ( $v$ leads $x$ )
Acceleration $a(t)$	$-\omega^2 A$ to $+\omega^2 A$	$\pi$ ( $a$ is opposite to $x$ )

Remember

All three graphs are sinusoidal in nature.
$v(t) = \omega A\cos\left(\omega t + \tfrac{\pi}{2}\right)$ . So the velocity leads the displacement by $\tfrac{\pi}{2}$ .
$a(t) = \omega^2 A\cos(\omega t + \pi)$ . So the acceleration is always opposite to the displacement.

2.8Force Law for SHM

Move a body a little away from its equilibrium (mean) position. A restoring force then acts on it, opposite to the displacement. It tries to bring the body back to equilibrium. For a small displacement, this force is proportional to the displacement. If the body is left free, it moves back towards the mean position.

On the way back, the body gains kinetic energy. So it does not stop at the mean position. It overshoots and goes to the other side. There, a restoring force again slows it down and brings it back. This keeps repeating.

Displaced→pulled back→gains KE and overshoots→other side→repeats

In this way, the restoring force makes the body oscillate about the mean position with a definite period.

Derivation

By Newton's second law, $F = ma$ . Put $a(t) = -\omega^2 x(t)$ and write the constant $m\omega^2$ as $k$ :

$F(t) = -m\omega^2 x(t) = -kx(t) \qquad \ldots\text{(i)}$

Force Law for SHM

F(t) = -kx(t)

where, $k = m\omega^2$ or $\omega = \sqrt{\dfrac{k}{m}}$

Definition

Alternative Definition of SHM

Eq. (i) gives another definition of SHM. Simple harmonic motion is the motion of a particle under a force that is proportional to its displacement and is always directed towards the mean position.

Part 3

Energy in SHM, Spring and Pendulum

A particle in SHM has both kinetic energy and potential energy. This part finds both energies. It then studies two real systems that move in SHM: a spring–block system and a simple pendulum.

3.1Kinetic Energy in SHM

To move a body away from its equilibrium position, we must do work on it. This work gives the body potential energy. When we release it, the body moves back with some velocity. So it gains kinetic energy.

Work done to displace→gains PE→released→moves back, gains KE

Note

In SHM, the kinetic energy and the potential energy both keep changing between zero and their maximum values.

Derivation

Kinetic Energy

At any instant, the displacement of the particle is $x = A\cos(\omega t + \phi)$ . So its velocity is:

$v = \dfrac{dx}{dt} = -\omega A\sin(\omega t + \phi)$
Kinetic energy of the particle at displacement $x$ :
$K = \dfrac{1}{2}mv^2 = \dfrac{1}{2}m\omega^2 A^2\sin^2(\omega t + \phi)$
Since $k = m\omega^2$ :
$K = \dfrac{1}{2}kA^2\sin^2(\omega t + \phi)$
Now use $\sin^2 = 1 - \cos^2$ and $x = A\cos(\omega t + \phi)$ :
$A^2\sin^2(\omega t + \phi) = A^2[1 - \cos^2(\omega t + \phi)]$
$= A^2 - A^2\cos^2(\omega t + \phi) = A^2 - x^2$
So, in terms of displacement:
$K = \dfrac{1}{2}m\omega^2(A^2 - x^2) = \dfrac{1}{2}k(A^2 - x^2)$

Kinetic Energy in SHM

K = \dfrac{1}{2}k(A^2 - x^2)

K

also changes periodically with time. It becomes zero when the displacement ismaximum

K

has its largest value at themean position

Period of the kinetic energy

T/2

3.2Potential Energy in SHM

Derivation

Potential Energy

Let the displacement of the particle from its equilibrium position be $x$ . The restoring force on it is $F = -kx$ .

Move the particle further by a tiny distance $dx$ . The work done by the restoring force is:
$dW = F\,dx = -kx\,dx$
Total work done by this force as the particle moves from the mean position ( $x = 0$ ) to $x$ :
$W = \displaystyle\int dW = -\int_0^x kx\,dx = -k\left[\dfrac{x^2}{2}\right]_0^x = -\dfrac{1}{2}kx^2$
The work done against the restoring force is $-W$ . It is stored as potential energy:
$U = -W = \dfrac{1}{2}kx^2 = \dfrac{1}{2}m\omega^2 x^2$

Potential Energy in SHM

U = \dfrac{1}{2}kx^2

Put $x = A\cos(\omega t + \phi)$ to write $U$ in terms of time:

U = \dfrac{1}{2}m\omega^2 A^2\cos^2(\omega t + \phi) = \dfrac{1}{2}kA^2\cos^2(\omega t + \phi)

U

is also periodic, with period

T/2

U

is zero at themean position

U

is maximum at anextreme position

Remember

KE and PE Behave Oppositely

At the mean position: KE is maximum, PE is zero.
At an extreme position: KE is zero, PE is maximum.
Both energies have the same period, $T/2$ .

3.3Total Energy in SHM

At any displacement $x$ , the total energy of the oscillating particle is:

E = K + U = \dfrac{1}{2}k(A^2 - x^2) + \dfrac{1}{2}kx^2

Total Energy in SHM

E = \dfrac{1}{2}kA^2

E = \dfrac{1}{2}m\omega^2 A^2 = 2\pi^2 m\nu^2 A^2 \qquad [\because \omega = 2\pi\nu]

Does Not Depend On

The total mechanical energy of an SHM does not depend on time or on displacement.

Depends On

It depends on the maximum displacement, that is, the amplitude.

Principle

Conservation of Energy

So, when there is no friction, the total energy of an SHM is conserved.

So the total energy $E$ varies directly as:

its mass

E \propto m

the square of the frequency

E \propto \nu^2

the square of the amplitude

E \propto A^2

3.4Graphs of Energy in SHM

At the Mean Position (

x = 0

)

(i) Kinetic energy, $K = \dfrac{1}{2}k(A^2 - 0^2) = \dfrac{1}{2}kA^2$

(ii) Potential energy, $U = \dfrac{1}{2}k(0^2) = 0$

All the mechanical energy is kinetic.

At the Extreme Positions (

x = \pm A

)

(i) Kinetic energy, $K = \dfrac{1}{2}k(A^2 - A^2) = 0$

(ii) Potential energy, $U = \dfrac{1}{2}kA^2$

All the energy is potential. The KE is zero.

Using these results, we can draw $K$ , $U$ and $E$ against the displacement $x$ and against the time $t$ :

K

U

and

E

against displacement

x

K

U

and

E

against time

t

(starting from

x = +A

)

PE or KE completes two vibrations while the SHM completes one (each repeats after

T/2

)2 : 1

So the frequency of PE or KE isdouble that of SHM

KeyThe total energy stays constant at every time $t$ and every displacement $x$ .

3.5Oscillations of a Spring

A pure SHM is possible only under some conditions. Two examples of pure SHM under such conditions are a spring–block system and a simple pendulum.

Take a massless spring on a frictionless horizontal table. One end is fixed to a rigid support. A block of mass $m$ is fixed to the other end. Pull the block to the right by a small distance $x$ and release it. The restoring force then makes it move back and forth about its equilibrium position.

Equilibrium, stretched by

x

and compressed by

x

. The force always points towards the mean position

F = -kx

Note

Force Constant

k

$k$ is the force constant of the spring. It is the restoring force per unit deformation of the spring. The minus sign shows that the force acts opposite to $x$ .

Derivation

Time Period of a Spring–Block System

Let $\dfrac{d^2x}{dt^2}$ be the acceleration of the block. By Newton's second law, $F = ma = m\dfrac{d^2x}{dt^2}$ . Equate this with the restoring force:

$m\dfrac{d^2x}{dt^2} = -kx$
$\dfrac{d^2x}{dt^2} = -\dfrac{k}{m}\,x = -\omega^2 x$
So the acceleration is proportional to $x$ and acts opposite to it. Hence the block moves in simple harmonic motion.
Its time period is:
$T = \dfrac{2\pi}{\omega} = \dfrac{2\pi}{\sqrt{k/m}}$

Time Period

T = 2\pi\sqrt{\dfrac{m}{k}}

Frequency

\nu = \dfrac{1}{T} = \dfrac{1}{2\pi}\sqrt{\dfrac{k}{m}}

A very stiff spring (large

k

) with a light body (small

m

)

T

small,

\nu

large

3.6Simple Pendulum

Definition

Ideal Simple Pendulum

An ideal simple pendulum is a point mass hung from a fixed support by an inextensible, weightless string. Such a pendulum cannot be made in practice.

In practice, we hang a small metal bob from a support with a long, fine cotton thread. This works as a simple pendulum.

A simple pendulum

iIdeal

Point mass + weightless string.

Not possible in practice.

iiPractical

Small metal bob + fine cotton thread.

The real-life version.

3.7Time Period of a Simple Pendulum

A small bob of mass $m$ hangs from an inextensible, massless string of length $L$ . At equilibrium, the bob is vertically below the point of suspension.

Displace the bob slightly to one side and release it. It oscillates about the mean position O. At some instant, let the bob be at A. Its displacement along the arc is $OA = x$ . The string makes an angle $\theta$ with the vertical.

Forces acting on the bob

iWeight

The weight $mg$ of the bob acts vertically downwards.

iiTension

The tension $T$ acts along the string, towards the support.

Two rectangular components of

mg

mg\cos\theta

— along the string

It is balanced by the tension $T$ in the string.

mg\sin\theta

— along the tangent

It gives a restoring torque. It tries to bring the bob back to the mean position.

Forces on the bob: weight

mg

and tension

T

mg

splits into

mg\cos\theta

and

mg\sin\theta

Derivation

Time Period of a Simple Pendulum

Let $L$ = length of the simple pendulum, $I$ = moment of inertia of the pendulum about the pivot, and $\alpha$ = angular acceleration about the pivot.

Restoring torque of $mg\sin\theta$ about the pivot. The minus sign shows that the torque acts to reduce $\theta$ :
$\tau = -(mg\sin\theta)L = -mgL\sin\theta \qquad \ldots\text{(i)}$
For rotation, the torque is also:
$\tau = I\alpha \qquad \ldots\text{(ii)}$
From Eqs. (i) and (ii):
$-mgL\sin\theta = I\alpha \;\Rightarrow\; \alpha = \dfrac{-mgL\sin\theta}{I}$
With $\theta$ in radian, $\sin\theta = \theta - \dfrac{\theta^3}{3!} + \dfrac{\theta^5}{5!} - \cdots$ For a small $\theta$ , $\sin\theta \approx \theta$ . So:
$\alpha = \dfrac{-mgL\,\theta}{I}$
This is the angular form of $a = -\omega^2 x$ . Comparing the two:
$\omega = \sqrt{\dfrac{mgL}{I}}$
Since $\omega = \dfrac{2\pi}{T}$ :
$\dfrac{2\pi}{T} = \sqrt{\dfrac{mgL}{I}} \;\Rightarrow\; T = 2\pi\sqrt{\dfrac{I}{mgL}} \qquad \ldots\text{(iii)}$
The whole mass of a simple pendulum is in the bob, taken as a point at distance $L$ from the pivot. So $I = mL^2$ . Put this in Eq. (iii):
$T = 2\pi\sqrt{\dfrac{mL^2}{mgL}} = 2\pi\sqrt{\dfrac{L}{g}}$

Note

Why Is It SHM?

The angular acceleration is proportional to the angular displacement $\theta$ , but it has the opposite sign. So when the bob moves to the right, the pull towards the left grows. The bob stops and comes back. The same thing happens on the left side. So the bob swings to and fro. Hence a simple pendulum swinging through small angles moves in approximately SHM.

Time Period of a Simple Pendulum

T = 2\pi\sqrt{\dfrac{L}{g}}

T

depends on the length

L

and the acceleration due to gravity

g

depends

T

does not depend on the mass $m$ of the bobindependent

T = 2\pi\sqrt{L/g}

holds only when

L

is small compared with the Earth's radiussmall

L

only

Important

If the Length Is Very Large

T = 2\pi\sqrt{\dfrac{R}{\left(1 + \tfrac{R}{L}\right)g}}, \qquad \text{where } R = \text{radius of the Earth}

\text{As } L \to \infty:\; T = 2\pi\sqrt{\dfrac{R}{g}} \qquad\quad \text{When } L = R:\; T = 2\pi\sqrt{\dfrac{R}{2g}}

3.8Quick Revision

Term	Key points
Periodic motion	Repeats itself after regular intervals of time.
Oscillatory motion	To-and-fro motion, again and again, about a fixed point (the mean position). Also called vibratory motion.
SHM	Periodic motion in which the restoring force is directly proportional to the displacement from the mean position: $F \propto x$ or $F = -kx$ . Here $k$ = force constant (spring constant).
Displacement	$x = A\cos(\omega t + \phi)$ , or in sine form $y = A\sin(\omega t + \phi)$ . $A$ = amplitude, $\omega$ = angular frequency $\left(\omega = \dfrac{2\pi}{T} = 2\pi\nu\right)$ , $\phi$ = initial phase.
Time period $T$	Time taken to complete one oscillation about the mean position.
Frequency $\nu$	Oscillations completed per second. $\nu = \dfrac{1}{T}$ , in hertz. $1\ \text{Hz} = 1$ oscillation per second $= 1\ \text{s}^{-1}$ .
Angular frequency $\omega$	Frequency multiplied by $2\pi$ : $\omega = 2\pi\nu = \dfrac{2\pi}{T}$ . SI unit: $\text{rad s}^{-1}$ .
Phase	Gives the state of the particle at any instant: its position and its direction of motion.
Velocity	$v = \omega\sqrt{A^2 - x^2}$ or $v(t) = -\omega A\sin(\omega t + \phi)$ . At the mean position ( $x = 0$ ): $v = A\omega = v_{max}$ (velocity amplitude). At $x = \pm A$ : $v = 0$ .
Acceleration	$a = -\omega^2 x$ or $a(t) = -\omega^2 A\cos(\omega t + \phi)$ . At $x = 0$ : $a = 0$ . At $x = \pm A$ : $a_{max} = \mp A\omega^2$ (acceleration amplitude $= A\omega^2$ ).
Time period in SHM	$T = \dfrac{2\pi}{\omega} = 2\pi\sqrt{\left\|\dfrac{x}{a}\right\|} = 2\pi\sqrt{\dfrac{\text{Displacement}}{\text{Acceleration}}}$
Energy in SHM	$U = \tfrac{1}{2}m\omega^2x^2 = \tfrac{1}{2}kx^2$ $K = \tfrac{1}{2}m\omega^2(A^2 - x^2) = \tfrac{1}{2}k(A^2 - x^2)$ $E = U + K = \tfrac{1}{2}m\omega^2A^2 = 2\pi^2m\nu^2A^2$
Simple pendulum	In practice: a heavy but small metal bob hung by a light, inextensible, flexible string. $T = 2\pi\sqrt{\dfrac{L}{g}}$ and $\nu = \dfrac{1}{2\pi}\sqrt{\dfrac{g}{L}}$ .

End of Chapter 13 · notesshala.in