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NDA Physics · Oscillations and Waves

Simple Harmonic Motion and General Waves

Simple harmonic motion is the to-and-fro motion produced when the restoring force is proportional to the displacement and points back toward the mean position; a wave is a disturbance that carries energy — and every wave reflects, carries energy and exerts pressure, but only light needs no medium.

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Why this matters

Six PYQs and the conceptual spine of the whole chapter. The bank leans on two HARD ideas — that two instants of a simple-harmonic motion are in the same phase only when they are a whole number of periods apart, and that sound and water waves (unlike light) cannot cross a vacuum. The easier marks test the defining property of SHM (force ∝ −displacement) and the basic relations ω = 2πf and 'motion repeats after every nT'. Get the definition of SHM and the shared-versus-unique wave properties watertight and the pendulum subtopic that follows is just one formula on top.

Concept 1 of 3

What makes a motion simple-harmonic

Intuition

Pull a mass on a spring aside and let go: the further you pull it, the harder it is pulled back, and the pull is always toward the rest point. That single rule — a restoring force that grows in proportion to how far you are displaced and always points home — is what makes a motion simple-harmonic. The result is a smooth, endlessly repeating swing.

Definition

A particle is in simple harmonic motion (SHM) when its restoring force (and hence its acceleration) is directly proportional to the displacement from the mean position and directed opposite to it — back toward the mean position.

  • Restoring force: F=−kxF = -kx (the minus sign means 'toward the mean position').
  • Acceleration: a=−ω2xa = -\omega^2 x, so acceleration is largest at the extremes and zero at the mean position.
  • The motion is periodic (it repeats in equal time intervals) and oscillates between +A+A and −A-A, the amplitude.

Defining condition of SHM

F=−kxa=−ω2xF = -kx \qquad a = -\omega^2 x
  • Frestoring force
  • xdisplacement from the mean position
  • kforce constant (positive)
  • ω\omegaangular frequency

Worked example

A block on a spring feels a restoring force F=−40xF = -40x newtons when displaced xx metres from rest. What is the force when the block is 0.2 m to the right of the mean position, and which way does it point?
Practice this conceptself-check · 4 quick reps

From the bank · past-year question

Example 1Oscillations and WavesEASY
Which one of the following statements is true for a simple harmonic oscillator ?

[Q93 · Apr · 2021]

Acceleration in SHM is NOT constant

A distractor claims the oscillator's acceleration is constant. It is not — a=−ω2xa = -\omega^2 x changes continuously, peaking at the extremes and vanishing at the mean position. Only the angular frequency ω\omega is constant.

The restoring force opposes the displacement

The force is proportional to displacement and in the OPPOSITE direction (toward the mean position). An option saying 'force in the same direction as displacement' describes an unstable push-away, not SHM.

Concept 2 of 3

Period, frequency, and phase

Intuition

The period T is the time for one complete swing; the frequency f is how many swings happen per second, so the two are reciprocals. The phase tells you where in the cycle the particle is. Because the motion repeats exactly every period, two instants look identical — same position and same direction of travel — only when they are a whole number of periods apart.

Definition

  • Period TT: the least time after which the motion repeats. The motion also repeats after 2T,3T,…2T, 3T, \dots — i.e. after every nTnT for positive integer nn.
  • Frequency f=1/Tf = 1/T: cycles per second (hertz).
  • Angular frequency ω=2πf=2π/T\omega = 2\pi f = 2\pi/T.
  • Phase: the stage of the cycle. Two instants t1t_1 and t2t_2 are in the same phase (identical displacement AND direction of motion) iff t2−t1=nTt_2 - t_1 = nT.

Period, frequency, angular frequency

f=1Tω=2πf=2πTf = \dfrac{1}{T} \qquad \omega = 2\pi f = \dfrac{2\pi}{T}
  • Tperiod (s)
  • ffrequency (Hz)
  • ω\omegaangular frequency (rad/s)
time tx+A−Aone period Tsame phase

The displacement repeats every period T. Two instants are in the same phase only when they are separated by a whole number of periods (Δt = nT).

Worked example

A particle in SHM has a period of 5 s. (a) What is its frequency? (b) At t = 2 s it is at a crest moving downward. List two later instants when it is again at a crest moving downward.
Practice this conceptself-check · 4 quick reps

From the bank · past-year question

Example 2Oscillations and WavesEASY
Which one of the following is the correct relation between frequency ff and angular frequency ω\omega ?

[Q120 · Apr · 2017]

Motion repeats after EVERY nT, not 'only once' after T

A favourite NDA distractor states 'the motion repeats after time T only once'. False — T is the LEAST repeat time, but the motion also repeats after 2T, 3T, … i.e. after every nT. The 'only once' wording is the wrong statement to pick when asked which is NOT correct.

Same phase needs Δt = nT — count whole periods

Reading a displacement-time graph, two instants are in phase only if their separation is an integer multiple of the period. Δt = T/2 gives equal displacement but OPPOSITE direction of motion — that is anti-phase, not the same phase.
Drill 2 more on period, frequency, and phase

Concept 3 of 3

Shared and unique properties of waves

Intuition

Sound, ripples on water and light look very different, but as waves they share a common toolkit: each one reflects, each one carries energy from place to place, and each one exerts a pressure on what it hits. The one property that splits them apart is the need for a medium — sound and water waves must travel through matter, while light (an electromagnetic wave) sails happily through empty space.

Definition

All waves — electromagnetic, sound, and water — share three properties:

  • Reflection: every wave bounces off a boundary.
  • Energy transport: every wave carries energy without carrying the medium along with it.
  • Pressure: every wave exerts a pressure (radiation pressure for light; wave/sound pressure for mechanical waves).

The dividing property: only electromagnetic waves travel through a vacuum. Sound and water waves are mechanical — they need a material medium and cannot cross empty space.

TransversecresttroughtravelLongitudinalcompressionrarefactiontravel

In a transverse wave the medium moves perpendicular to travel (crests and troughs); in a longitudinal wave it moves parallel (compressions and rarefactions). Both carry energy without carrying matter.

Worked example

An astronaut on the airless Moon claps two stones together a few metres from a companion. Will the companion hear the clap? Will the companion see the stones meet? Explain.
Practice this conceptself-check · 4 quick reps

From the bank · past-year question

Example 3Oscillations and WavesHARD
Which of the following statements about electromagnetic waves, sound waves and water waves is/are correct? 1. They exhibit reflection. 2. They carry energy. 3. They exert pressure. 4. They can travel in vacuum. Select the correct answer using the code given below:

[Q119 · Apr · 2018]

Sound and water waves cannot cross a vacuum — only light can

The statement 'they can travel in vacuum' is true for electromagnetic waves but FALSE for sound and water waves. When a question groups all three wave types, the vacuum-travel option must be excluded — it is the trap that turns '1, 2, 3 and 4' into the wrong answer.

Lightning before thunder = light is faster, not 'sound is slower than expected'

Seeing a flash before hearing the thunder shows light (~3×10⁸ m/s) outruns sound (~343 m/s) over the same distance. The conclusion is about the SPEED difference, not about the brightness or the intensity of the flash.
Drill 1 more on shared and unique properties of waves

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Watch out for (6)

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6 questions from the bank — paginated, with cart and Word-export support.

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