📘 Oscillations 

Oscillation is the repeated to-and-fro motion of a particle about its mean or equilibrium position.

Examples:

• Oscillation of a simple pendulum

• Vibrations of a tuning fork

• Vibrations of a stretched string

• Vibrations of molecules 

The periodic to-and-fro motion of a particle about a fixed mean position is called oscillatory motion.

Important terms

Mean position: Equilibrium position of the particle.

Extreme position: Position at maximum displacement from mean position.

Amplitude AA: Maximum displacement from the mean position. 

Particle executes SHM when its acceleration is directly proportional to its displacement from the mean position and is always directed towards the mean position.

a∝−x 

a=−ω2x​

where:

• a = acceleration

• x = displacement

• ω = angular frequency 

Restoring Force

F=ma

Therefore,

F=−mω2x

or

F=−kx

The negative sign indicates that the restoring force is directed towards the mean position.

The displacement of a particle executing SHM can be written as

x=Asin⁡(ωt+ϕ)

or

x=Acos⁡(ωt+ϕ)

where:

• A = Amplitude

• ω = angular frequency

• t = time

• ϕ = initial phase

Angular Frequency

ω = 2πf 

Differential Equation of SHM 

SHM as Projection of Uniform Circular Motion 

SHM can be represented as the projection of a particle executing uniform circular motion on any diameter of the circle.

If a particle moves around a circle with angular velocity ω\omega, its projection on a diameter executes SHM.

This method is useful for understanding:

• displacement

• velocity

• acceleration

• phase

• phase difference 

Phase 

The quantity

ωt+ϕ

is called the phase of the oscillating particle.

Phase tells us the instantaneous state of oscillation.

Phase Difference

For two oscillations,

Δϕ=ϕ2−ϕ1

For one complete oscillation:

Δϕ=2π​ 

Energy in SHM 

Simple Pendulum 

Year-wise Distribution 

Topic-wise Frequency & Recurring Patterns 

Differential Equation of Linear S.H.M. & Derivations (a, v, x) — (Highest Yield: 3–4 Marks)

• Trend: The single most consistent core derivation in the chapter. It appears as an entire 4-mark Section D question or a 3-mark Section C question across both old and new syllabus cycles.

• Core Questions:

  • "State the differential equation of linear S.H.M. Hence, obtain an expression for acceleration, velocity, and displacement of a particle performing linear S.H.M."
    (March 2018, March 2019, Feb 2023, March 2026)
    
    

  • "Define linear S.H.M. Obtain the differential equation of linear S.H.M." (Feb 2020) 

Differential Equation of Linear S.H.M. & Derivations (a, v, x) — (Highest Yield: 3–4 Marks)

• Trend: The single most consistent core derivation in the chapter. It appears as an entire 4-mark Section D question or a 3-mark Section C question across both old and new syllabus cycles.

• Core Questions:

  • "State the differential equation of linear S.H.M. Hence, obtain an expression for acceleration, velocity, and displacement of a particle performing linear S.H.M."
    (March 2018, March 2019, Feb 2023, March 2026)
    
    

  • "Define linear S.H.M. Obtain the differential equation of linear S.H.M." (Feb 2020)

Composition of Two Linear S.H.M.s (1–4 Marks)

• Trend: Tested either as an extensive analytical derivation in Section D or as short amplitude calculation MCQs in Section A. 

Kinematics of S.H.M. & Max Velocity / Max Acceleration Numericals (1–2 Marks) 

• Define Simple Harmonic Motion (SHM). (Repeated)
  
  

• Define time period and frequency. (2019, 2022)
  
  

• Define amplitude of oscillation. (2020, 2023)
  
  

• Write expression for angular frequency. (2021) 

• State conditions for SHM. (2018, 2022)
  
  

• Write equation of SHM and explain terms. (2020, 2023)
  
  

• Obtain relation between frequency and time period. (2019)

• Derive expression for velocity in SHM. (2021, 2023)
  
  

• Derive expression for acceleration in SHM. (2019, 2022)
  
  

• Energy distribution in SHM. (2020) 

• Derivation of time period of simple pendulum. (Repeated: 2018, 2021, 2024)
  
  

• Derivation of time period of spring–mass system. (2019, 2022)
  
  

• Total energy in SHM derivation. (2020, 2023) 

1. Time period of simple pendulum calculation.
   
   

2. Spring–mass system time period.
   
   

3. Maximum velocity and acceleration in SHM.
   
   

4. Total energy of oscillating particle.
   
   

5. Displacement, velocity, acceleration using SHM equation.
   
   

6. Effect of change in length of pendulum on time period.
   
   

7. Calculation of frequency or angular frequency. 

✅ Time period of simple pendulum derivation 

✅ Spring-mass system numericals 

✅ Energy in SHM 

✅ Velocity & acceleration equations 

✅ Numerical on amplitude and frequency 

• 1 derivation + 1 numerical is almost guaranteed

• PV diagrams = easy 2–3 marks

• Thermodynamics alone gives 10–14 marks 

• Write formula → substitution → answer with unit 

• Always draw neat PV diagrams 

• Remember sign convention (very important!)