Introduction to Simple Harmonic Motion
We find motion in everything in nature. The simple harmonic motion characterizes that type of motion which is oscillating. A simple example is the vibration of guitar strings. We can see a number of back-and-forth motions in the string before coming to rest. This motion is actually called periodic motion. Thus simple harmonic motion (SHM) is also a basic motion seen around us, whose analysis and mathematical treatment is critical.
Simple Harmonic Motion is a periodic to-and-fro motion in which a body moves on a straight line about a fixed point (called the mean or equilibrium point). The motion continuously repeats after a certain interval of time known as the time period. What makes SHM special is that the restoring force acting on the body is always directed toward the equilibrium position and is directly proportional to the displacement from that position.
SHM is the basis for understanding waves, alternating current (AC), musical instruments, molecular vibrations, atomic structure, and even the large-scale oscillations of planets and stars. In physics, whenever a system is slightly disturbed from its stable position, it tends to oscillate. These oscillations often approximate SHM if the disturbance is small. Thus, studying SHM gives us a powerful tool to understand many physical phenomena.
Characteristics of Simple Harmonic Motion
A simple harmonic motion has following features:
(i) The motion must be oscillatory
The body must move repeatedly back and forth around a fixed mean position. This motion should continue over time.
(ii) The restoring force must follow Hooke’s Law
The restoring force (or acceleration) must always be directed toward the equilibrium and must be proportional to displacement from it.
Mathematically, F∝−x
or,
F=−kx [Equation 1]
where:
- F = restoring force
- x = displacement from mean position
- k = force constant
The negative sign shows that the force acts opposite to the displacement. Equation (1) also shows that force is in linear relation with displacement.
Some examples of systems that approximately satisfy these conditions are:
- Mass attached to an ideal spring
- Simple pendulum (for small angles)
- Small oscillations of a loaded beam
- Vibrations of tuning forks
- Suspension springs in vehicles
Mathematical Description of SHM
Let us consider a particle moving in SHM along a straight line ( let along x-axis). Let, x=0 be the mean position. Now, if the body is oscillated, the following mathematical treatment is done.
General Expression of Displacement
The displacement x at time t of the particle is given by:
x(t)=Asin(ωt+ϕ) [Equation 2]
or
x(t)=Acos(ωt+ϕ) [Equation 3]
Where:
- A = amplitude (maximum displacement)
- ω = angular frequency (rate of oscillation)
- φ = phase constant or initial phase
- t = time
Both sine and cosine forms describe the same motion but depend on the starting position and phase.
Amplitude (A)
The amplitude is the maximum distance the body moves on either side of the equilibrium. It determines how “far” the oscillation reaches but does not affect the time period.
Angular Frequency (ω)
Angular frequency represents how fast the body oscillates. Angular frequency ω and time period T are related as:
ω = 2πf = 2π/T [Equation 4]
Phase (φ)
The phase tells us the initial state of the particle and at what position it reaches after time t. It also tells from where the motion begins (mean position, extreme position, or between them). SHM is periodic and repeats after every time period:
T=2π/ω
Differential Equation of SHM
The restoring force in SHM follows:
F = −kx
Using Newton’s second law F = ma:
ma = −kx
Or, a = -k/m x
Acceleration is proportional to displacement and is in opposing nature with the displacement. Since:
a = d^2x/dt^2
We get the differential equation:
d^2x/dt^2+ω^2x=0 [Equation 5]
where
ω = √k/m
This is the fundamental equation of SHM. Solving this differential equation gives:
x(t)=Asin(ωt+ϕ)
which we introduced earlier.
Thus, SHM is the solution of the simplest form of second-order differential equation. It involves harmonic oscillations.
Energy in Simple Harmonic Motion
In simple harmonic motion, kinetic and potential energy vary continuously. However, the total energy is always conserved.
(i) Potential Energy (PE)
When the body vibrates, work is done against the restoring force and the potential energy is given by:
PE = 1/2kx^2 [Equation 6]
(ii) Kinetic Energy (KE)
As the body crosses the mean position, the velocity becomes maximum and hence has maximum kinetic energy which is given by:
KE = 1/2mv^2 [Equation 7]
(iii) Total Energy (E)
The total mechanical energy in SHM is:
E=1/2kA^2 [Equation 8]
Energy Distribution
At extreme positions (x = ±A):
PE is maximum
KE is minimum (zero)
At mean position (x = 0):
KE is maximum
PE is minimum (zero)
This continuous exchange between kinetic energy and potential energy give rise to the oscillatory motion.
Phase, Amplitude, Frequency and Period
To understand SHM deeply, we must understand its characteristic quantities.
(i) Amplitude
It is the maximum displacement from equilibrium position of the particle.
(ii) Frequency (f)
It is the number of oscillations taken by the particle per second.
f=1/T
(iii) Time Period (T)
In SHM time taken to complete one full oscillation is called time period.
T=2π/ω
(iv) Angular Frequency (ω)
ω=2πf
(v) Phase (φ)
It gives the position of the body at any instant during the oscillation.
(vi) Phase Difference
For two particles in SHM making an angle Δϕ with each other, two conditions arise:
- If Δϕ = 0, they move together (same phase).
- If Δϕ = π they move opposite (completely out of phase).
Equation for Velocity and Acceleration
We have from displacement equation:
x = Asin(ωt+ϕ)
To obtain the velocity:
Velocity is the rate of change of displacement:
v = dx/dt
Velocity is maximum at the mean position and zero at the extreme positions.
vmax = Aω [Equation 9]
Maximum velocity:
To obtain the acceleration (a)
Acceleration is obtained by:
a = d^2x/dt^2
= −Aω^2sin(ωt+ϕ) [Equation 10]
Acceleration is maximum at the extreme positions and zero at the mean position.
Maximum acceleration:
amax = Aω^2 [Equation 11]
Relationship Between a and x
a =−ω^2x [Equation 12]
This relationship is unique to SHM and is the exact mathematical condition that defines it.
Superposition of SHMs
As mentioned above, the displacement in SHM is given by the equation (1). Superposition means the addition of two or more than two quantities. If the SHMs are of same frequency but different phases and amplitudes, the length of a vector is calculated by the cosine rule (c^2 = a^2 + b^2 – 2abcosC).
Similarly, if they are in the opposite direction but have the same amplitude, we use the trigonometric relation: cosA + cosB = 2cos((A+ B)/2) cos((A- B)/2).
The oscillation has an angular frequency of ω1+ω2/2, followed by frequency of ω1-ω2/2. When the original frequencies are very near, the modifying frequency becomes very slow. We hear “beats” with a frequency of ω1-ω2 (here we don’t use ½ times, because node comes twice in each cycle).
Simple Pendulum as SHM
A simple pendulum contains a heavy mass suspended at the bottom called a bob. The suspension is made by a light string which is inextensible. Now, if it is displaced from the mean position, it keeps oscillating back and forth until it comes to rest. This resembles a SHM.
Why is a pendulum SHM?
For small angles (<15°), restoring force is proportional to displacement:
F ≈ −mgθ
Since θ∝x, the force is proportional to displacement.
Time Period of a Simple Pendulum
T = 2π√l/g [Equation 13]
Where:
- l = length of the string
- g = acceleration due to gravity
Notes:
- The time period has nothing to do with the mass of the bob.
- Time period increases if length increases.
- Time period decreases if gravitational acceleration increases.
This is why pendulums run slower at high altitudes (where g is smaller).
Mass–Spring System as SHM
A mass attached to a spring is the most common example of SHM.
Case 1: Horizontal Spring System
If a system of spring is stretched or compressed by a mass, the spring exerts a restoring force to come back to its mean position. This force is given by,
F=−kx
The equation of motion will be:
a = −k/mx
ω = √k/m
T=2π√m/k [Equation 14]
Case 2: Vertical Spring System
In vertical motion, gravity shifts the equilibrium position, but the oscillations are still SHM with the same time period as the horizontal case.
Note:
- Stiffer springs (larger k) oscillate faster.
- Heavier masses oscillate slower.
- Amplitude does not affect time period.
Mass-spring systems are widely used in:
- Shock absorbers
- Vehicle suspensions
- Measuring instruments like spring balances
Damped and Driven Harmonic Motion
The two types of harmonic motion are:
(i) Damped Harmonic Motion
Damping means the amplitude of vibration is gradually decreased with time. This decrease happens due to the presence of some force called the damping force. This may be due to friction or the air resistance. This force causes the loss of energy of the motion and hence energy dissipates until the system comes to rest.
Types of damping:
- Light damping: This damping occurs when oscillations continue but decrease slowly. After multiple decreases in oscillation, the system comes to rest. For example, car suspension.
- Critical damping: In this damping, the system comes to rest or its mean position without oscillation showing any oscillation. For example, seismometers.
- Heavy damping: In this case no oscillation can be obtained but the system comes to equilibrium slowly as compared to critical damping. For example, swing coming to rest.
(ii) Forced or Driven Oscillations
In contrast to damped harmonic oscillations, the driven oscillations are the type of oscillations which are made by giving external force to the system. Thus, the system oscillates in a frequency different from its natural frequency. The resonance is likely to occur if the driving frequency meets the natural frequency of the system.
Resonance
A phenomenon where amplitude becomes very large when driving frequency = natural frequency.
Examples:
- Breaking of a bridge due to soldiers marching
- Resonance in musical instruments (guitar body, violin)
- Tuning radio to a specific frequency
Applications of Simple Harmonic Motion in Daily Life
SHM is not just a topic of physics; it appears practically everywhere.
(i) In Clocks
In pendulum clocks and quartz watches, time measurement is done accurately by implying periodic oscillations.
(ii) Musical Instruments
Guitar strings, tuning forks all vibrate in SHM producing sound waves.
(iii) Engineering
- Shock absorbers
- Vehicle suspension systems
- Building design to reduce earthquake damage
(iv) Electronics
AC circuits behave like SHM and have their own natural frequencies.
(v) Atomic and Molecular Vibrations
Atoms in solids oscillate around their mean positions in an approximate SHM pattern, leading to:
- Heat capacity
- Thermal vibrations
- Acoustic properties
(vi) Medical Technology
SHM principles are used in ultrasound imaging and MRI resonance techniques.
(vii) Communication Technology
In radio transmitters and receivers, electronic circuits are practically the oscillating circuits implying SHM.
Conclusion
Simple Harmonic Motion (SHM) is actually the harmony of nature. From musical instruments to complex microscopic phenomena like molecular vibration, follow simple harmonic motion. The periodic vibration of the body back and forth, from mean position to maximum amplitude is described by this concept. Thus we can study how an object restores its initial position after the disturbance and how energy transforms between kinetic and potential forms. SHM is the basic term to form the foundation of various natural and artificial phenomena.
The core meaning of simple harmonic motion also describes the symmetry of nature and how every disturbed object tries to maintain this symmetry. Symmetry is a great topic not only in simple mechanics but also in particle physics, nuclear physics, solid state physics and quantum mechanics. Thus, it is the most basic form of the periodic motion required to understand other complex phenomena.
References
Learning, L. (2021). Simple Harmonic Motion: A Special Periodic Motion. Fundamentals of Heat, Light & Sound.
Motion, S. H. (2006). Simple Harmonic Motion. A A, 4(2), 2.
https://en.wikipedia.org/wiki/Simple_harmonic_motion
https://byjus.com/jee/simple-harmonic-motion-shm
https://www.geeksforgeeks.org/physics/simple-harmonic-motion
