Have you ever wondered about the kind of motion of the pendulum of a wall clock? Why does the bungee jumper moves to and fro at the end of the jump? These categories of motion are termed Periodic or Simple Harmonic Motion. We are, in fact, surrounded by objects that perform periodic motion.

Simple Harmonic Motion or SHM is a specific type of periodic motion that is very easy to understand and reproduce mathematically, and many of the periodic motions that we see in our day to day lives can be modelled as SHM. In this article, we will provide detailed information on Simple Harmonic Motion. Continue reading to find out more!

Periodic Motion

Periodic motion is a type of motion that repeats itself (in terms of position, velocity, and acceleration of the moving body) after a constant interval of time.

Example: The pendulum oscillates between two extremes with a mean position. It moves left extreme to mean position and then from mean position to right extreme and then back to left extreme passing through the mean point.

A particle moving in a circle or executing a circular motion comes back to the initial position after some interval of time.

Oscillatory Motion

The periodic “to and fro” motion of a particle about a point (known as equilibrium or mean position) is known as Oscillatory Motion or simply Oscillation. For a particle to execute oscillatory motion, there should be a force acting on it that should always try to bring the particle back to its equilibrium or mean position. This force is known as restoring force. For a force to be a restoring force, its direction should always be opposite to that of the displacement of the particle from its mean position.

Simple Harmonic Motion

Simple Harmonic Motion or SHM is a specific type of oscillation in which the restoring force is directly proportional to the displacement of the particle from the mean position.
\(F ∝ – x\)
\(F = – Kx\)
Here,
\(F\) is the restoring force.
\(x\) is the displacement of the particle from the mean position.
\(K\) is the force constant.
Thus the acceleration of the particle is given by,
\(a = \frac{F}{m}\)
\(a = \frac{-Kx}{m}\)
Where,
\(m\) is the mass of the particle.
Let,
\(ω^2 = \frac{K}{m}\)
As, \(\frac{?}{?}\) is a positive constant.
\(⇒ a = -ω^2 x\)
We know that acceleration in terms of velocity and displacement is given by,
\(a = v \frac{dv}{dx}\)
\( \Rightarrow v\frac{{dv}}{{dx}} = – {\omega ^2}x\)
\( \Rightarrow \int_{{v_0}}^v {vdv} = – \int_{{x_0}}^x {{\omega ^2}xdx} \)
\( \Rightarrow \frac{{{v^2} – {v_o}^2}}{2} = – {\omega ^2}\frac{{\left( {{x^2} – {x_0}^2} \right)}}{2}\)
\( \Rightarrow {v^2} = {\omega ^2}\left[ {{{\left( {\frac{{{v_0}}}{\omega }} \right)}^2} + x_0^2 – {x^2}} \right]\)
Let,
\(A = \sqrt {{{\left( {\frac{{{v_0}}}{\omega }} \right)}^2} + {x_0}^2} \)
\( \Rightarrow {v^2} = {\omega ^2}\left[ {{A^2} – {x^2}} \right]\)
\( \Rightarrow v = \omega \sqrt {{A^2} – {x^2}} \)
We know that velocity in terms of displacement is given by,
\(v = \frac{{dx}}{{dt}}\)
\( \Rightarrow \int_{{x_0}}^x {\frac{1}{{\sqrt {{A^2} – {x^2}} }}dx} = \mathop \smallint \nolimits_0^t \omega dt\)
\( \Rightarrow \left[ {{\text{si}}{{\text{n}}^{ – 1}}\left( {\frac{x}{A}} \right)} \right]_{{x_0}}^x = \omega t\)
On solving, we get the general equation for a Simple Harmonic Motion,
\( ⇒ x = A \sin (ωt + δ)\)
In differential, form we have,
\(\frac{{{d^2}x}}{{d{t^2}}} + {\omega ^2}x = 0\)
If the mean position does not lies on the origin, then we have,
\(\frac{{{d^2}x}}{{d{t^2}}} + {\omega ^2}x = C\)
Where mean position lies at,
\({x_0} = \frac{C}{{{\omega ^2}}}\)

Simple Harmonic Motion: Important Terms

Mean Position: The position at which the net force on the particle in SHM is zero is known as the mean position. Velocity at the mean position is maximum, whereas acceleration is minimum.

Extreme Position: The Position located at the distance equal to the amplitude of the SHM is known as extreme position. The velocity at extreme position is zero while the acceleration is maximum.
Amplitude of the SHM: The maximum displacement from the mean position is known as amplitude.
Phase: Phase or status of the SHM is a quantity which is inside of the trigonometric function for position of the particle. It defines the state which is, the position and the direction of motion of the SHM.
If the equation of the SHM is,
\(x = A \sin (ωt + δ)\)
The phase of the SHM will be the common solution of,
\(x = A \sin (\phi)\)
And,
\(v = A ω \cos (\phi)\)
Angular Frequency: For a SHM given by equation,
\(x = A \sin (ωt + δ)\)
\(ω\) is known as the angular frequency.
Time Period: The time interval after which the particle comes in same phase.
Linear Frequency: The inverse of the time period is known as frequency.

Relation Between SHM and Circular Motion

Simple Harmonic motion can be represented as the projection of uniform circular motion with an angular frequency of the SHM is equal to the Angular velocity.
Simple Harmonic Motion will be the motion of the shadow of the particle when light rays parallel to the plane of the motion is incident on the particle.

Uniform Circular Motion	Simple Harmonic Motion
Radius	Amplitude
Angular position with respect to reference line at \(t = 0\)	Phase constant
Angular Velocity	Angular frequency
Angular position with respect to reference line at any time \(t\)	Phase

Energy in SHM

FIG
Kinetic energy is given by,
\(KE = \frac{1}{2}m{v^2}\)
We know that for SHM the velocity of the particle is given by,
\(v = \omega \sqrt {{A^2} – {x^2}} \)
Thus, the kinetic energy will be given by,
\( \Rightarrow KE = \frac{1}{2}m{\left( {\omega \sqrt {{A^2} – {x^2}} } \right)^2} = \frac{1}{2}m{\omega ^2}\left( {{A^2} – {x^2}} \right)\)
Change in potential energy is given by,
\(U\left( x \right) – U\left( 0 \right) = – \left( {\mathop \smallint \nolimits_0^x Kxdx} \right)\)
If the potential at mean position is zero then,
\(U\left( x \right) = \frac{1}{2}K{x^2} = \frac{1}{2}m{\omega ^2}{x^2}\)
Total energy is the sum of kinetic and potential energy and is constant,
\(E\left( x \right) = KE\left( x \right) + U\left( x \right)\)
\(E\left( x \right) = \frac{1}{2}m{\omega ^2}{A^2}\)

Kinetic Energy is maximum at the mean position while potential energy is maximum at the extreme position while the total energy remains constant.

Simple Pendulum

When a particle of some mass (known as bob) is hanged with the help of a string from a point, this arrangement is known as a simple pendulum. When the bob is displaced by some angle, then the pendulum starts the periodic motion, and for a small value of angle of displacement, the periodic motion is simple harmonic motion with the angular displacement of the bob.
\(F = mg \sin (θ)\)
\(a = g \sin (θ)\)
For small oscillation, \(θ\) will be small,
\({\text{sin}}\left( \theta \right) = \theta = \frac{x}{l}\)
\( \Rightarrow a = g \theta\)
\( \Rightarrow a = g\frac{x}{l}\)
Thus the angular frequency is given by,
\( \Rightarrow {\omega ^2} = \frac{g}{l}\)
The time period of the pendulum is given by,
\(T = \frac{{2\pi }}{\omega }\)
\( \Rightarrow T = 2\pi \sqrt {\frac{l}{g}} \)

Physical Pendulum and Angular SHM

For simple pendulum we consider the mass of the string to be negligible as compared to the Bob, but for physical pendulum the mass of the string need not be negligible. In fact, any rigid body can act as a physical pendulum.
BY the definition of the centre of mass, the total mass of the body can be considered to be at the centre of mass. By writing the torque equation for the rigid body about the fixed point, we get the angular acceleration of the rigid body is directly proportional to the angular displacement by using small angle approximation.
Time period of a physical pendulum is given by,
\(T = 2\pi \sqrt {\frac{{{I_0}}}{{mg{l_{cm}}}}} \)
Where,
\(I_0\) is the moment of inertia about the fixed point trough which the axis passes.
\(l_{cm}\) is the distance of the centre of mass from the axis point.

Combination of SHMs

A particle can undergo more than one simple harmonic motion.
Along the same direction,
A particle is performing a combination of two SHM in the same direction.
\({x_1} = {A_1}{\text{sin}}\left( {\omega t} \right)\)
\({x_2} = {A_2}{\text{sin}}\left( {\omega t + \delta } \right)\)
\(x\left( t \right) = {x_1}\left( t \right) + {x_2}\left( t \right)\)
On solving we get,
\(x(t) = (A_1 + A_2 \cos(δ)) \sin(ωt) + A_2 \sin(δ) \cos(ωt)\)
Let,
\({A_1} + {A_2}{\text{cos}}\left( \delta \right) = A\,{\text{cos}}\left( \phi \right)\)
\({A_2}\,{\text{sin}}\left( \delta \right) = A\,{\text{sin}}\left( \phi \right)\)
\( \Rightarrow A = \sqrt {{A_1}^2 + {A_2}^2 + 2{A_1}{A_2}{\text{cos}}\left( \delta \right)} \)
\({\text{tan}}\left( \phi \right) = \frac{{{A_2}{\text{sin}}\left( \delta \right)}}{{{A_1} + {A_2}{\text{cos}}\left( \delta \right)}}\)

Along the perpendicular direction,
The motion of a particle under two SHMs which are in the direction perpendicular to each other.

\(x = {A_1}{\text{sin}}\left( {\omega t} \right)\)
\(y = {A_2}{\text{sin}}\left( {\omega t + \delta } \right)\)
Expanding \(y = {A_2}{\text{sin}}\left( {\omega t + \delta } \right)\) and substituting the value of \({\text{sin}}\left( {\omega t} \right)\) from \(x = {A_1}{\text{sin}}\left( {\omega t} \right)\),
\( \Rightarrow \frac{y}{{{A_2}}} = \frac{x}{{{A_1}}}{\text{cos}}\left( \delta \right) + \sqrt {1 – {\text{si}}{{\text{n}}^2}\left( {\omega t} \right){\text{sin}}\left( \delta \right)}\)
On further simplification we get,
\( \Rightarrow {\left( {\frac{y}{{{A_2}}}} \right)^2} – 2\frac{y}{{{A_2}}}\frac{x}{{{A_1}}}{\text{cos}}\left( \delta \right) + \frac{x}{{{A_1}}} = {\text{si}}{{\text{n}}^2}\left( \delta \right)\)
For a different value of the \(\delta\), the particle will have a different path,

1. If \(\delta = 0\), Particle follows a straight line.

2. If \(\delta = \pi \), Particle follows a straight line.

3. If \(\delta = \frac{\pi }{2}\), Particle follows an elliptical path,

4. If the amplitude of both the SHMs is equal, the resultant path will be a circle.

Applications of Simple Harmonic Motion

Below we have provided some of the applications of SHM:

1. It is used in wall clocks.
2. It is used in the back and forth movement of a cradle.
3. It is used in the back and forth movement of swings in the parks.
4. Simple Harmonic Motion is used for bungee jumping.
5. SHM is used as a Car Shock Absorber.

Solved Examples of Simple Harmonic Motion

Q.1.Calculate the time period of the block of mass ‘\(m\)’ attached to a spring of the spring constant ‘\(k\)’, when displaced by a distance of ‘\(x\)’ \(\rm{m}\) from the equilibrium position.

Ans: Given,
Mass of the block is \(m\).
Spring constant of the spring is \(k\).
Step 1: Find the equilibrium position.
For equilibrium position the total force on the block will be zero.
\(mg = kx_0\)
Step 2: Displace the particle from the equilibrium position by distance \(x\).

Step 3: Make a body diagram and find the acceleration of the particle. Determine the value of angular frequency and find the time period.

\(F = mg – k (x + x_0)\)
\(F = -kx\)
\(⇒ ω^2 = \frac{k}{m}\)
Thus the time period is given by,
\( \Rightarrow T = 2\pi \sqrt {\frac{m}{k}} \)

Q.2. In the previous example, find the time period using the energy method.
Ans: Given,
Mass of the block is \(m\).
Spring constant of the spring is \(k\).
Step 1: Find the equilibrium position.
For equilibrium position, the total force on the block will be zero.

\(mg = kx_0\)
Step 2: Write the expression for total energy of the system as function of position.
\(E = f(x) = \rm{const}\)
\( \Rightarrow {{E}} = \frac{1}{2}{{m}}{{{v}}^2} + \frac{1}{2}{{k}}{\left( {{{x}} + {{{x}}_0}} \right)^2} – {{mgx}} = {\text{const}}\)
Step 3: Differentiating with respect to time and simplifying.
\(\frac{{d{{E}}}}{{d{{t}}}} = \frac{1}{2}{{m}} \times 2{{v}}\frac{{d{{v}}}}{{d{{t}}}} + \frac{1}{2}{{k}} \times 2\left( {{{x}} + {{{x}}_0}} \right)\frac{{d{{x}}}}{{d{{t}}}} – {{mg}}\frac{{d{{x}}}}{{d{{t}}}}\)
On solving, we get,
\( \Rightarrow {{a}} = – \frac{{{k}}}{{{m}}}\)
\( \Rightarrow {\omega ^2} = \frac{k}{m}\)
Thus, the time period is given by,
\(T = 2\pi \sqrt {\frac{m}{k}} \)

Summary

1. A body performs SHM when the force on the body is directly proportional to the displacement but in the opposite direction of the displacement.
2. At the mean position the acceleration of the particle is zero and the velocity of the particle is maximum.
3. At the extreme position the acceleration of the particle is maximum while the velocity is zero.
4. Every Simple harmonic motion can be represented as the projection of uniform circular motion.
5. Total energy of the particle performing SHM remains constant.
\(E\left( x \right) = KE\left( x \right) + U\left( x \right)\)
\(E\left( x \right) = \frac{1}{2}m{\omega ^2}{A^2}\)
6. The time period of the simple pendulum is given by,
\(T = \frac{{2\pi }}{\omega }\)
\( \Rightarrow T = 2\pi \sqrt {\frac{l}{g}} \)

FAQs on Simple Harmonic Motion

Q.1. What is periodic motion?
Ans: Periodic motion is a type of motion in which the particle repeats the set path after some interval of time. Example: Earth revolves around the earth.

Q.2. Where does the magnitude acceleration have the maximum value?
Ans: Magnitude of acceleration is maximum at both extremes, which is at the distance equal to the amplitude of the SHM from the mean position.

Q.3. What happens to the energy of the particle in SHM?
Ans: The total energy of the particle remains constant while the kinetic energy is maximum at the mean position while the potential energy is maximum at extremes.

Q.4. What is the expression for the time period of the pendulum?
Ans: The time period of the simple pendulum is given by,
\(T = \frac{{2\pi }}{\omega }\)
\( \Rightarrow T = 2\pi \sqrt {\frac{l}{g}} .\)

Q.5. Is acceleration constant during SHM?
Ans: No, acceleration is directly proportional to the displacement, but it is in the opposite direction to the displacement.

Q.6. What are some of the applications of SHM?
Ans: The applications of SHM are as follows:
1. It is used in wall clocks.
2. It is used in the back and forth movement of a cradle.

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