Oscillations and Simple Harmonic Motion (SHM)

1.0What is an Oscillation? 

An oscillation is a repetitive motion of an object about a central, or equilibrium, position. Think of a swing moving back and forth, a guitar string vibrating, or a clock pendulum ticking. These are all examples of oscillations.

The key characteristic of an oscillation is its periodicity—the motion repeats itself over a regular interval of time. The path of an oscillating object doesn't necessarily have to be a straight line. For example, a planet orbiting the sun is a periodic motion, but it's not typically considered an oscillation in the simple sense, as it doesn't move about a fixed equilibrium point in a back-and-forth manner.

2.0Periodic Motion vs. Oscillatory Motion

(a) Periodic Motion: Any motion which repeats itself after a regular interval of time is called periodic motion or harmonic motion.

The constant interval of time after which the motion is repeated is called time period.

Ex: (i) Motion of Planets around the sun

(ii) Motion of the pendulum

(b) Oscillatory Motion: The motion of a body is said to be oscillatory or vibratory motion if it moves back and forth (to and fro) about a fixed point after a certain interval of time.
The fixed point about which the body oscillates is called mean position or equilibrium position.

Ex : (i) Vibration of the wire of "Sitar". 

(ii) Oscillation of the mass suspended from spring.

3.0What is Simple Harmonic Motion (SHM)

Simple Harmonic Motion (SHM) is the most fundamental and simplest type of oscillation. It's a special case of oscillatory motion where the restoring force is directly proportional to the displacement and acts in the opposite direction.

S.H.M. are of two types

(a) Linear S.H.M.
When a particle moves to and fro about a fixed point (called equilibrium position) along a straight line then its motion is called linear simple harmonic motion.
Example : Motion of a mass connected to spring.

(b) Angular S.H.M.
When a system oscillates angularly with respect to a fixed axis then its motion is called angular simple harmonic motion.
Example :- Motion of a bob of simple pendulum.

Necessary Conditions to execute S.H.M.

• Motion of particle should be oscillatory.
• Total mechanical energy of particle should be conserved (Kinetic energy + Potential energy = constant)
• Extreme position should be well defined.
• In linear S.H.M the restoring force (or acceleration) acting on the particle should always be proportional to the displacement of the particle and directed towards the equilibrium position.
∴ F ∝ −x or a ∝ −x
Negative sign shows that direction of force and acceleration is towards equilibrium position and x is displacement of particle from equilibrium position.
• In angular S.H.M. the restoring torque (or angular acceleration) acting on the particle should always be proportional to the angular displacement of the particle and directed towards the equilibrium position.
∴ τ ∝ −θ or α ∝ −θ

4.0Differential equation of SHM and Graphs

Characteristics of SHM

Understanding these parameters is vital for solving questions on SHM.

• Displacement (x): The instantaneous position of the oscillating particle with respect to its mean position.

Displacement in SHM

(i) The displacement of a particle executing linear S.H.M. at any instant is defined as the distance of the particle from the mean position at that instant.

(ii) It can be given by relation x = A sin ωt or x = A cos ωt. The first relation is valid when the time is measured from the mean position and the second relation is valid when the time is measured from the extreme position of the particle executing S.H.M. along a straight-line path.

(iii) Maximum value of displacement is = ± A

• Amplitude (A): The maximum displacement of the particle from its mean position. It's the maximum value of x.
• Time Period (T): The time taken to complete one full oscillation. For a spring-mass system, $T=2\pi \sqrt{\frac{m}{k}}$. For a simple pendulum, $T=2\pi \sqrt{\frac{L}{g}}$
• Frequency (ν or f): The number of oscillations completed per unit time. It is the reciprocal of the time period (f=1/T). The SI unit is Hertz (Hz).
• Angular Frequency (ω): The rate of change of phase angle in radians per unit time. It is related to frequency and time period by $\omega =2\pi \nu =\frac{2\pi }{T}$

5.0Displacement, Velocity, and Acceleration in SHM

Let's derive the equations for these key variables. Consider a particle performing SHM along the x-axis.

Displacement: The general equation for displacement is given by:

x(t)=Asin(ωt+ϕ)

or

x(t)=Acos(ωt+ϕ)

• A is the amplitude.
• ω is the angular frequency.
• ϕ is the initial phase angle or phase constant, which depends on the initial conditions of the motion.

Velocity: Velocity is the time derivative of displacement (v=dtdx​).

If x(t)=Asin(ωt+ϕ), then:

v(t)=Aωcos(ωt+ϕ)

The maximum velocity is $v_{max}=A\omega ,$ which occurs at the equilibrium position (x = 0).

Acceleration: Acceleration is the time derivative of velocity (a = dv/dt).

a(t) = −Aω² sin(ωt + φ)

This can be rewritten using the displacement equation as:

a(t) = −ω² x(t)

The maximum acceleration is $a_{max}=A{\omega }^2$, which occurs at the extreme positions (x = ±A).

6.0Graphical Representation

Graphical study of displacement, velocity, acceleration and force in S.H.M.

S. No.	Graph	In form of t	In form of x	Maximum value
1.		x = A sin ωt	x = x	x = ±A
2.		v = Aω cos ωt	v = ±ω√(A² − x²)	v = ±ωA
3.		a = −ω²A sin ωt	a = −ω²x	a = ±ω²A
4.		F = −mω²A sin ωt	F = −mω²x	F = ±mω²A

7.0The Simple Pendulum: A Classic Example of SHM

A simple pendulum consists of a point mass (bob) suspended from a fixed support by a light, inextensible string.

For small angular displacements (θ < 10°), the restoring force component is approximately proportional to the displacement (F ≈ −mgθ = −mg x/L). This leads to SHM.

The time period of a simple pendulum is:

T = 2π √(L/g)

• L is the length of the string.
• g is the acceleration due to gravity.

8.0Spring-Mass System: A Real-World SHM Example

A spring-mass system is another perfect example of SHM. A block of mass m is attached to a spring with spring constant k and oscillates on a frictionless horizontal surface.

The restoring force is the spring force, F=−kx.

The time period of this system is:

$T=2\pi \sqrt{\frac{m}{k}}$

• m is the mass of the block.
• k is the spring constant.

Energy in SHM

The total mechanical energy of a particle undergoing SHM remains constant if there is no damping. This energy is the sum of its kinetic energy (KE) and potential energy (PE).

Kinetic Energy (KE):

$KE=\frac{1}{2}m{v}^2=\frac{1}{2}m{\omega }^2{A}^2{\cos }^2(\omega t+\varphi )$

The maximum KE occurs at the mean position (x=0).

Potential Energy (PE): The PE is stored in the system due to the restoring force (e.g., in the spring).

$PE=\frac{1}{2}k{x}^2=\frac{1}{2}k{A}^2{\sin }^2(\omega t+\varphi )$

The maximum PE occurs at the extreme positions (x=±A).

Total Energy (E): $E=KE+PE=\frac{1}{2}m{\omega }^2{A}^2{\cos }^2(\omega t+\varphi )+\frac{1}{2}k{A}^2{\sin }^2(\omega t+\varphi )$

Since ${\omega }^2$=k/m or $m{\omega }^2$=k, we get:

$$\begin{gathered} E=\frac{1}{2}k{A}^2({\cos }^2(\omega t+\varphi )+{\sin }^2(\omega t+\varphi )) \\ E=\frac{1}{2}k{A}^2=\frac{1}{2}m{\omega }^2{A}^2 \end{gathered}$$

The total energy in SHM is directly proportional to the square of the amplitude ($A^2$).

9.0Applications of SHM

SHM principles are applied in various real-world systems:

• Pendulum Clocks: Utilize the periodic motion of a pendulum to keep accurate time.
• Mass-Spring Systems: Demonstrate basic SHM principles and are used in mechanical oscillators.
• Vibrations in Structures: Understanding SHM helps in designing buildings and bridges to withstand oscillatory forces.
• Musical Instruments: Strings and air columns vibrate in SHM to produce sound.

Damped and Forced Oscillations

While ideal SHM assumes no energy loss, real-world systems experience damping, where energy is gradually lost, usually due to friction or air resistance.

• Damped Oscillations: The amplitude of oscillation decreases exponentially over time. The motion is no longer truly periodic. A common example is a car's shock absorber.

• Forced Oscillations: An external periodic force is applied to the oscillating system. The system eventually oscillates at the frequency of the external force.

• Resonance: This is a special and important case of forced oscillation. If the frequency of the external force matches the natural frequency of the oscillating system, the amplitude of oscillations increases drastically. This principle is used in musical instruments and radio receivers but can be destructive in bridges and buildings.