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SHM & Oscillations

Reduce the motion to ẍ + ω²x = 0 (or θ̈ + ω²θ = 0) in 15–35 seconds; everything else—T, v_max, energy, phase—follows from that ω.

Portion MechanicsFocus Clean marks from ω identification, energy share, phase and light damping/resonance ideasTarget 60–90sPortion share 32.3%Trend Falling share
Linear SHM means acceleration ∝ −displacement from equilibrium. Hooke’s law F=−kx is the spring prototype. Angular SHM uses the same math with θ. Forced and damped 1D oscillations add damping and a driving frequency; resonance is maximum response near the natural frequency (exact peak depends on what is plotted). SHM template: find equilibrium → displace → linearise F or τ → read ω → match IC for A,φ. If linearisation fails, it is not exam SHM.

How Advanced asks this

  • Find T or ω from F=−k_eff x after shifting to equilibrium
  • Spring combinations, elevators, or vertical spring mean position
  • Energy, amplitude, phase difference between x and v
  • Simple/physical pendulum; liquid U-tube; floating body SHM
  • Two-SHM superposition qualitative (same line)
  • Damped/forced: qualitative amplitude decay, resonance peak statements

Traps that leak marks

  • Measuring spring PE from unstretched length while using SHM x from equilibrium without adjusting
  • Using pendulum formula for large amplitudes without caveat
  • Putting g into mass–spring period
  • Saying force is always toward mean but using wrong mean
  • Assuming energy conserved in clearly damped motion
  • Claiming resonance exactly at ω₀ for every plotted response without reading the stem
  • Adding periods instead of combining k_eff correctly
  • Using T=2π√(l/g) for a physical pendulum without k_eff
  • Calling motion SHM when restoring is not linear
  • Mixing amplitude with instantaneous displacement in energy

01. Standard form: x = A sin(ωt+φ) or A cos(ωt+φ). ω=√(k_eff/m) for mass–spring equivalents; T=2π/ω.

02. Energy: E=½kA²=½mω²A²; K and U exchange; at x, v=ω√(A²−x²).

03. Angular SHM: small-angle pendulum θ̈+(g/L)θ=0 ⇒ ω=√(g/L). Physical pendulum: ω=√(mgd/I).

04. Damped: amplitude decays; underdamped oscillates with slightly shifted ω. Forced: steady amplitude depends on drive frequency; resonance when drive is near natural frequency for the observed quantity.

05. SHM means a = −ω²x along the chosen displacement from equilibrium. Find equilibrium first (net force/torque zero), then expand restoring force for small x—ω comes from that linear coefficient.

06. Energy in SHM is ½kA²=½kx²+½mv² (or torsional analogue). Phase and initial conditions fix A and φ; do not paste A=x_max from a non-equilibrium start without solving.

07. Physical/simple pendulum and floating-body SHM all reduce to the same template: restoring ∝ −displacement near equilibrium. Large angles break the sinθ≈θ model.

Official · 2026

Syllabus for this topic

Official syllabus points for SHM & Oscillations only — not the full Mechanics portion map. Portion hub: Mechanics · Mechanics analysis.

Syllabus on this page is topic-scoped: SHM & Oscillations. The parent portion (Mechanics) also covers other topics — use the portion hub for the full map.

Must cover (this topic)

  • Forced/damped oscillation & resonance

Gaps / watch-outs

No topic-specific exclusions flagged — finish the unit points below for SHM.

Source: https://jeeadv.ac.in/documents/jee-advanced-2026-syllabus.pdf

Unit 01

SHM & Oscillations

high

Exam focusLinear/angular SHM, damped/forced oscillation, and resonance.

Syllabus points

  • Forced and damped oscillation (1D), resonance; linear and angular SHM

Full Physics syllabus →

Cross-topic stems

Mixed joins for this topic

Advanced rarely asks a chapter in isolation. These are tagged multi-idea Physics stems that touch SHM & Oscillations — practise the knowledge bridge, not only the single-chapter drill.

Physics2026 · Paper 1 · Q2single-correct

Mutual inductance + LC resonance

EMI – Mutual inductance+AC / LC circuits+Effective inductance

Knowledge bridge

Step 1: read the coupling M = L/2 and series inductance to form L_eff. Step 2: replace the network by one equivalent L_eff (here 3L/4). Step 3: apply ω₀ = 1/√(L_eff C) to get 2/√(3LC).

Why it feels hard

Students remember ω=1/√(LC) but forget to rebuild L_eff from mutual terms.

What to practise

For every coupled-inductor problem, compute L_eff first, then resonate.

2026 year review →Physics notes →Quick card →Syllabus →

All mixed joins →

Why methods matter

Common methods that change your score

Casting the equation as ẍ+ω²x=0 yields T, A and energy without solving a full DE.

01

Equilibrium shift first

Constant forces only shift the mean; ω still √(k/m).

Do thisFor vertical springs or extra constant forces, find new equilibrium, then displace from there.

Why it works

Constant forces only shift the mean; ω still √(k/m).

See it once

mg extends spring by δ=mg/k; measure x from that point ⇒ same T as horizontal.

02

k_eff assembly

Converts cluttered spring diagrams into a single SHM constant.

Do thisSeries/parallel springs → one k_eff; then ω=√(k_eff/m).

Why it works

Converts cluttered spring diagrams into a single SHM constant.

See it once

Two springs parallel: k_eff=k₁+k₂; series: 1/k_eff=1/k₁+1/k₂ (same force).

03

Force-law linearisation

Shows when motion is approximately SHM and gives ω without memorising every formula.

Do thisFor pendulum, floating body, U-tube: expand restoring force to first order in displacement.

Why it works

Shows when motion is approximately SHM and gives ω without memorising every formula.

See it once

U-tube: restoring pressure ⇒ a = −(2g/L)x with L total liquid length.

04

Phase portrait check

Kills wrong multi-correct statements about simultaneous maxima.

Do thisRemember x and v are π/2 out of phase; a and x are π out of phase.

Why it works

Kills wrong multi-correct statements about simultaneous maxima.

See it once

When |x|=A, v=0 and |a|=ω²A; when x=0, |v|=ωA and a=0.

Processed examples

Sample questions · full working

Read the impact line first. Then notice which family/model was chosen. The solution is written the way your rough sheet should look in the hall.

01

bankFamily · Standard SHM identification

Sample question

A mass 0.25 kg on a spring k=100 N/m oscillates. Find the period.

This stem rewards early recognition of the standard shm identification model.

Model chosenT=2π√(m/k).

Processed solution

  1. T=2π√(0.25/100)=2π×0.05=π/10 s.

Answerπ/10 s

Takeaway: ω=√(k/m) identifies the SHM clock.

02

standardFamily · Energy in SHM

Sample question

A particle in SHM has amplitude 4 cm and maximum speed 40 cm/s. Find the maximum acceleration.

This stem rewards early recognition of the energy in shm model.

Model chosenω=v_max/A; a_max=ω²A.

Processed solution

  1. ω=40/4=10 s⁻¹.
  2. a_max=100×4=400 cm/s².

Answer400 cm/s²

Takeaway: Amplitude and v_max fix ω without needing m or k.

03

twistFamily · Physical / torsional / angular SHM

Sample question

A uniform rod of length ℓ and mass m is pivoted at one end and displaced by a small angle. Find the period of small oscillations.

This Advanced-level variation couples hidden constraints with the physical / torsional / angular shm model.

Model chosenτ=−mg(ℓ/2)θ and I=mℓ²/3 about the pivot.

Processed solution

  1. Restoring torque ≈ −(mgℓ/2)θ for small θ.
  2. I_pivot=mℓ²/3.
  3. θ̈ + (mgℓ/2)/(mℓ²/3) θ = 0 ⇒ ω²=3g/(2ℓ).
  4. T=2π√(2ℓ/(3g)).

Answer2π√(2ℓ/(3g))

Takeaway: Angular SHM needs both the torque constant and the correct I.

Reusable models

Common question families

Advanced stems cluster. Name the family in 10–15s, write the fast model, then run the same path. Do not invent a new plan each time.

01

Standard SHM identification

Looks likex = A sin(ωt+φ) from force or energy

Fast modelReduce to ẍ + ω²x = 0; read ω.

Same solve path

  1. Write F or effective restoring torque.
  2. Cast as ẍ = −ω²x (or θ̈ = −ω²θ).
  3. Identify A from energy or IC.
  4. Answer T=2π/ω, v_max=Aω, etc.

Rough skeleton

  • restoring law
  • ω
  • A/T

Why faster

Recognition fixes the model before arithmetic begins.

Refuse this

Do not start blind expansion or casework before identifying the governing model.

02

Energy in SHM

Looks likeK/U exchange, amplitude from E, or mean values

Fast modelE = ½kA² = ½mω²A²; K+U=E at every point.

Same solve path

  1. Write total E.
  2. Express K(x) and U(x).
  3. Solve for A, v, or x.
  4. Use time averages only if asked.

Rough skeleton

  • E
  • K/U split
  • result

Why faster

Recognition fixes the model before arithmetic begins.

Refuse this

Do not start blind expansion or casework before identifying the governing model.

03

Physical / torsional / angular SHM

Looks likePendulum, rod, or torsional constant κ

Fast modelSmall-angle: τ = −κθ ⇒ ω=√(κ/I).

Same solve path

  1. Identify restoring torque law.
  2. Find I about pivot.
  3. Form ω=√(κ/I) or √(mgl/I) etc.
  4. Linearize sinθ≈θ if needed.

Rough skeleton

  • τ law
  • I
  • ω

Why faster

Recognition fixes the model before arithmetic begins.

Refuse this

Do not start blind expansion or casework before identifying the governing model.

04

Damped / forced / resonance sketch

Looks likeQualitative amplitude–frequency or decay

Fast modelName under/critical/overdamped; resonance near ω₀ for light damping.

Same solve path

  1. Identify driving and damping presence.
  2. State the qualitative regime.
  3. Use ω_res ≈ ω₀ for weak damping if numerical.
  4. Avoid full DE solution unless required.

Rough skeleton

  • regime
  • ω₀
  • qualitative/result

Why faster

Recognition fixes the model before arithmetic begins.

Refuse this

Do not start blind expansion or casework before identifying the governing model.

Hall timing

60–90s process

  1. 0–15s
    Identify equilibrium; write net restoring force/torque for small displacement.

    Underline given data, unknowns and constraints. Name the mechanical model before recalling any formula; discard relations whose assumptions are already broken.

  2. 15–35s
    Cast as m ẍ = −k_eff x or I θ̈ = −κ θ; read ω=√(k_eff/m).

    Draw axes, free-body or energy/impulse diagram. Write governing equations symbolically with consistent sign and reference.

  3. 35–75s
    Apply A, φ from IC; use energy or v=ω√(A²−x²) as needed.

    Solve algebraically as far as possible; substitute numbers only once. Keep vectors, components and units on the rough sheet.

  4. 75–90s
    Check small-angle validity, mean position vs amplitude, and damping/drive regime if present.

    Check dimensions, limiting cases and physical sense. For multi-correct, test every option independently rather than forcing one narrative.

After the model is locked

Quick formula bank

SHM definition

a = −ω² x; T=2π/ω; f=ω/(2π)

Linear SHM about equilibrium

Mass–spring

ω=√(k/m); T=2π√(m/k)

Horizontal or vertical ideal spring (about eq.)

Velocity in SHM

v = ω √(A² − x²); v_max = ωA

Given position in oscillation

Energy

E = ½ k A² = ½ m ω² A²; U=½ k x² (from mean)

Mechanical energy in ideal SHM

Simple pendulum

T=2π√(L/g) (small θ)

Point mass, light string

Angular SHM

θ = θ₀ sin(ωt+φ); ω=√(κ/I)

Restoring torque −κθ

Forced (qualitative)

steady amplitude large when ω_drive ≈ ω_natural (light damping)

Resonance discussion

Mental bank

  • a=−ω²x defines SHM
  • T=2π√(m/k) spring; T=2π√(L/g) simple pendulum
  • v=ω√(A²−x²)
  • E=½kA²; max K at mean
  • Vertical spring: same ω about new equilibrium
  • x and v phase difference π/2
  • Small angle needed for pendulum SHM
  • Resonance: drive near natural frequency (light damping)
  • Equilibrium first, then a=−ω²x
  • ω² = k_eff/m or κ/I for the linearised restoring law
  • E=½mω²A² shared by KE+PE
  • Time period independent of A only in true SHM

Make it automatic

Practice · checklist

Daily

  • Two k_eff / vertical-spring ω finds and one energy-at-x numerical.
  • One pendulum/U-tube linearisation and one phase multi-correct drill.

Weekly

  • One mixed SHM set including light damping/forced qualitative statements.
  • Derive U-tube and floating-body T once from F=ma.

100% checklist

  • Equilibrium identified and displacement measured from it
  • Equation cast as ẍ+ω²x=0 (or angular analogue)
  • ω from k_eff/m or κ/I
  • Small-angle or linearity assumption checked
  • Energy reference consistent with x origin
  • Phase relations between x,v,a correct
  • Damping/forcing treated qualitatively when exact ODE not required
  • Equilibrium identified and displacement measured from it
  • Linear restoring law derived or stated
  • ω, A, energy consistent with initial conditions
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