Oscillations & Simple Harmonic Motion
Understand how restoring force, phase and energy create back-and-forth motion, then apply SHM formulas to springs, pendulums and JEE Main problems.
The central idea of this chapter
Any motion that repeats itself can be treated as an oscillation. When the restoring force is proportional to displacement and opposite in direction, the motion becomes simple harmonic.
What should you understand first?
Oscillations introduces periodic motion, phase, and the special case of SHM, then applies these ideas to spring– mass systems, pendulums and energy graphs.
Core concepts
- Periodic and oscillatory motion High
- Definition of SHM High
- Phase and phase constant High
- Spring–mass system High
- Simple pendulum (small angle) Medium
SHM at a glance
- Displacement x = A sin(ωt + φ)
- Velocity v = ω√(A² − x²)
- Acceleration a = −ω²x
- Time period T = 2π/ω
- Total energy ½mω²A²
Oscillations and SHM formula sheet
For SHM, acceleration is always directed towards the mean position and proportional to displacement, which fixes the time period in terms of system parameters.
| Topic | Formula / Relation | Meaning or use |
|---|---|---|
| Equation of SHM | x(t) = A sin(ωt + φ) | A is amplitude, ω is angular frequency, φ is phase constant |
| Velocity in SHM | v = Aω cos(ωt + φ) = ω√(A² − x²) | Speed is maximum at mean position, zero at extremes |
| Acceleration in SHM | a = −ω²x | Restoring acceleration proportional to displacement |
| Time period (general) | T = 2π/ω | Time for one complete oscillation |
| Spring–mass (horizontal) | ω = √(k/m), T = 2π√(m/k) | k is spring constant, m is mass |
| Simple pendulum | T = 2π√(L/g) | Small-angle oscillations about mean position |
| Energy in SHM | E = ½mω²A² (constant) | Total mechanical energy independent of time |
| Instantaneous KE, PE | KE = ½mω²(A² − x²), PE = ½mω²x² | Energy shuttles between KE and PE, sum fixed |
How to read an oscillations question
Decide whether the problem wants time, position, phase or energy, then select the SHM relation that directly links those quantities.
Identify the quantity
- Time period or frequency? Check system (spring / pendulum)
- Displacement at time t? Use x(t) with phase
- Speed or acceleration? Use v, a relations
- Energy or amplitude? Use ½mω²A²
- Fraction of time in a region? Use phase angles
Quick reading checklist
- Is motion truly SHM? Check F ∝ −x
- Amplitude given or to be found? Watch units
- Phase reference? sin vs cos form
- Direction of motion? Sign of v, a
- Small-angle condition? Pendulum, springs with g
How to prepare Oscillations
Build intuition with graphs, then practise formula-based questions for time period, phase and energy in SHM.
What to do
- Start with definition of periodic motion and SHM
- Practise writing x, v and a in terms of phase
- Memorise time period results for spring and pendulum
- Draw energy vs displacement graphs to see KE/PE exchange
- Solve PYQs on phase, time fraction and graphical SHM
Common mistakes
- Assuming every oscillatory motion is SHM
- Forgetting that SHM time period is amplitude-independent
- Mixing up angular frequency ω with linear frequency f
- Using pendulum formula for large-angle motion
- Ignoring sign of displacement and direction of velocity
Ready to practise SHM?
Revise the Oscillations formula sheet, then attempt mixed JEE Main questions on springs, pendulums and energy in SHM.
Oscillations and SHM FAQ
Short answers to frequently tested JEE Main ideas.
Simple harmonic motion is a periodic motion in which the restoring force is directly proportional to displacement and directed towards the mean position, so acceleration is a = −ω²x.
The time period of SHM depends on system parameters like mass and spring constant for a spring–mass system, or length and g for a simple pendulum, but does not depend on amplitude for small oscillations.
In SHM, kinetic and potential energy keep exchanging while total mechanical energy remains constant, equal to ½mω²A².