Oscillations — NEET Notes

Periodic & oscillatory motion

A motion that repeats itself at regular intervals of time is called periodic motion. The orbit of Mercury (88 earth days), the appearance of Halley's comet (76 years), and the vibration of a quartz crystal (microseconds) are all periodic — they recur identically after a fixed interval. Within this family, a special subset moves to and fro about a mean position: a pendulum bob, a piston in a steam engine, a boat tossing on a river, a violin string. These are oscillatory motions. NCERT states the relationship sharply: every oscillatory motion is periodic, but every periodic motion need not be oscillatory. Uniform circular motion is periodic but not oscillatory — there is no mean position about which a body moves back and forth.

The period T is the smallest interval after which the motion repeats; its SI unit is the second. Its reciprocal is the frequency ν, the number of repetitions per unit time, measured in hertz (Hz). Thus ν = 1/T. The distinction between "oscillation" and "vibration" is one of convention rather than physics: low-frequency motion (a swaying tree branch) is called an oscillation, high-frequency motion (a guitar string) a vibration. Both obey the same mathematical machinery.

Within an oscillating system there is an equilibrium position — a point at which no net force acts on the body. Displace the body slightly and a restoring force appears, pulling it back. Left to itself in a frictionless world, the body would swing back and forth forever. In the real world, friction and viscosity dissipate energy and the motion eventually dies — unless an external periodic force keeps pumping energy in. We will return to damped and forced oscillations at the end of the chapter.

The simplest displacement function that captures periodic motion is the sinusoid: f(t) = A cos ωt, or equivalently A sin ωt. Adding sine and cosine of the same frequency produces another sinusoid, just with a different phase. A remarkable theorem proved by the French mathematician Joseph Fourier (1768–1830) states that any periodic function can be decomposed into a sum of sines and cosines. This is why the simple harmonic motion described below — the most basic of all sinusoidal oscillations — is also the building block of every more complicated periodic motion in physics.

Simple harmonic motion

Of all the oscillations one can construct, the simplest — and the one nature returns to over and over — is simple harmonic motion (SHM). NCERT defines it through its displacement function. A particle moving along an x-axis between the limits +A and −A is said to execute SHM if:

x(t) = A cos(ωt + φ)

The defining equation of SHM

Three quantities define the motion. The amplitude A is the magnitude of maximum displacement — the particle never strays farther than A from the centre, on either side. The angular frequency ω determines how fast the motion proceeds: it is related to the period by ω = 2π/T and to the frequency by ω = 2πν. The phase constant φ records where the particle is at t = 0; the full quantity (ωt + φ) is the phase of the motion. Two SHMs may share A and ω but differ in φ — they will trace identical cosine curves displaced sideways in time.

NIOS gives an equivalent definition via the force: a particle executes SHM if it moves to and fro about a fixed point under the action of a force F directly proportional to the displacement x from the fixed point and directed opposite to it — F = −kx. Either definition produces the other. The displacement equation differentiates twice to yield acceleration; multiplying by mass produces the force law. Both are equivalent statements of the same physical idea.

SHM & uniform circular motion

One of the most elegant results in this chapter is the geometric link between SHM and uniform circular motion. Imagine a particle P moving uniformly on a circle of radius A with angular speed ω. Now project P onto a diameter — say, the x-axis. As P circles, its projection P′ moves back and forth along the diameter. NCERT proves that the position of P′ is exactly x(t) = A cos(ωt + φ), which is the defining equation of SHM.

So SHM is the projection of uniform circular motion on a diameter of the reference circle. The radius of the circle equals the amplitude of the SHM. The angular speed of the circular motion equals the angular frequency of the SHM. The projection on a perpendicular diameter — say, the y-axis — gives a second SHM of the same amplitude but differing in phase by π/2. The geometry collapses what otherwise looks like a calculus problem into a single picture.

This reference-circle method is also the cleanest way to derive the formulas for velocity and acceleration. The speed of P in uniform circular motion is ωA, directed tangentially. Project that tangent vector onto the diameter, and you get the velocity of P′: v(t) = −ωA sin(ωt + φ). Centripetal acceleration of P is ω²A, directed inward along the radius. Project that, and you get a(t) = −ω²A cos(ωt + φ) = −ω²x(t). The kinematics of SHM falls out of one rotating arrow.

Velocity & acceleration in SHM

Starting from x(t) = A cos(ωt + φ) and differentiating, we obtain:

v(t) = −ωA sin(ωt + φ)    a(t) = −ω²A cos(ωt + φ) = −ω²x

Velocity and acceleration of a particle in SHM

All three quantities — position, velocity, acceleration — oscillate sinusoidally with the same period T, but their maxima differ and their phases differ. The displacement varies between −A and +A; the velocity between −ωA and +ωA; the acceleration between −ω²A and +ω²A. Velocity is maximum at the mean position (where x = 0) and zero at the extremes (where x = ±A). Acceleration is zero at the mean position and maximum at the extremes. They are out of step.

The phase differences are sharp and exam-friendly. With respect to displacement, the velocity has a phase lead of π/2 — it gets to its peak a quarter period earlier than x does. With respect to displacement, the acceleration has a phase difference of π — it is exactly out of phase. When x is positive, a is negative; when x is negative, a is positive. The acceleration is always directed back towards the mean position. This is the geometric meaning of a = −ω²x: the minus sign is a restoring direction, the ω² is a stiffness measure.

A useful corollary, often pulled by NEET, is the relation between speed and displacement: v = ω√(A² − x²). This follows from energy conservation, and lets you solve problems where the particle is at a particular intermediate displacement without bothering with phase angles. NEET 2017 set the magnitude of velocity equal to the magnitude of acceleration at a known x and asked for the time period — a one-line application.

Force law for SHM

Newton's second law converts the kinematic statement a = −ω²x into a statement about force. Multiply both sides by mass:

F(t) = ma(t) = −mω² x(t) = −k x(t),  with k = mω²

The force law for SHM

This is the single most important equation in the chapter. It tells you that any system whose restoring force is linear in displacement executes simple harmonic motion. The constant k — called the spring constant, force constant, or simply the "stiffness" — has units of N/m. Inverting the relation k = mω² gives ω = √(k/m), and therefore the period:

The force F = −kx is sometimes called the restoring force because it always points back towards the mean position, undoing whatever displacement you imposed. A particle oscillating under such a force is called a linear harmonic oscillator. If the force contains additional terms in x², x³, or higher powers — as it does for many real systems at large amplitudes — the motion is no longer strictly simple harmonic; the oscillator is "non-linear". But for small amplitudes, almost every restoring force in physics is approximately linear, which is why SHM is such a universal idealisation.

Energy in SHM

Two energy reservoirs trade back and forth in SHM. The kinetic energy K = ½ mv² depends on speed; the potential energy U = ½ kx² is stored against the spring. Their sum — the total mechanical energy — stays constant, as it must for motion under a conservative restoring force.

Plugging in v(t) = −ωA sin(ωt + φ):

K(t) = ½ mω²A² sin²(ωt + φ) = ½ kA² sin²(ωt + φ)

And plugging x(t) = A cos(ωt + φ) into U = ½ kx²:

U(t) = ½ kA² cos²(ωt + φ)

Adding the two and using sin²θ + cos²θ = 1:

E = K + U = ½ kA²  = constant

Total energy of a simple harmonic oscillator

Three implications matter for NEET. First, total energy is proportional to A² — doubling the amplitude quadruples the stored energy. Second, both K and U are always non-negative; one rises as the other falls; both peak twice in each period of x. This is why a particle in SHM oscillating with frequency n has its kinetic and potential energies oscillating at frequency 2n. NEET 2021 asked exactly this question. Third, at the mean position (x = 0) all the energy is kinetic; at the extremes (x = ±A) all the energy is potential. In between, the two trade smoothly via the identity sin² + cos² = 1.

A worked example fixes the numbers. Take a 1 kg block on a spring with k = 50 N/m, pulled to amplitude A = 10 cm and released. The total energy is ½ × 50 × (0.1)² = 0.25 J. At x = 5 cm, U = ½ × 50 × (0.05)² = 0.0625 J, leaving K = 0.25 − 0.0625 = 0.1875 J. The speed there is v = √(2K/m) = √(0.375) ≈ 0.61 m/s. The conservation works out to four decimal places.

The simple pendulum

A simple pendulum is an idealisation: a point-mass bob of mass m at the end of a massless, inextensible string of length L, swinging in a vertical plane under gravity. Galileo discovered its periodicity in his teens by timing the swing of a cathedral chandelier against his pulse. The crucial step in deriving its period is to recognise that, for small angular displacements θ, the restoring torque is approximately linear in θ — making the motion simple harmonic.

Three properties of the pendulum period repay memorisation. It is independent of mass. Because both the restoring force (mg sin θ) and the inertia (mL²) scale with mass, m cancels — a brick and a feather, hung from identical strings, swing in identical time. It is independent of amplitude, in the small-angle limit. NCERT tabulates sin θ versus θ and shows that even at 20° the approximation is good to about 2%. It depends on g: on the moon, where g = 1.7 m/s² (against earth's 9.8), a pendulum of the same length swings √(9.8/1.7) ≈ 2.4 times slower. NCERT Exercise 13.15 tests this directly.

NEET 2022 asked when two pendulums of length 121 cm and 100 cm, starting in phase at their mean position, will next be in phase. Their periods are in the ratio √(1.21/1.00) = 1.1, so after every 11 swings of the shorter and 10 swings of the longer they meet again. After 11 swings of the shorter pendulum, they are once more in phase — a clean application of T ∝ √L.

Spring-mass system

The other canonical SHM system is a block of mass m attached to a spring of force constant k, free to slide on a frictionless surface. Stretch the spring by x and Hooke's law gives a restoring force F = −kx. By the force law derivation, the block executes SHM with angular frequency ω = √(k/m) and period T = 2π√(m/k). NCERT and NIOS both prove this for the horizontal case; NIOS adds that for vertical oscillations the period is the same — gravity merely shifts the equilibrium position and does not change k or m.

The spring constant itself obeys algebra that NEET loves to test. For a spring of natural length L cut into pieces of lengths L₁ and L₂, the constants of the pieces are k₁ = kL/L₁ and k₂ = kL/L₂ — a shorter spring is stiffer, because k ∝ 1/L. For springs combined, the rules invert: series combinations give 1/k_eff = 1/k₁ + 1/k₂ (so two equal springs in series give half the stiffness, doubling the period); parallel combinations give k_eff = k₁ + k₂ (doubling the stiffness, shortening the period). NEET 2017 cut a spring into the ratio 1:2:3, combined the pieces in series and then in parallel, and asked for the ratio of effective constants — answer 1:11.

A useful family of SHM systems all share the same algebraic skeleton: an inertial parameter divided by a restoring-force parameter, all under a square root, all multiplied by 2π. The pendulum, the spring, a torsional oscillator (where a disc twists on a wire), even an LC circuit, all fit this template. The factor-grid below pulls them together.

Damped, forced & resonance

The SHM of NCERT's mathematical idealisation is a frictionless world. Real oscillators live in air, water, or other viscous media; some energy bleeds away every cycle as heat. The amplitude of a real pendulum or a real spring-mass system therefore decreases with time — exponentially, for small damping. These are damped oscillations. NIOS illustrates them with a metal block suspended from a spring inside a beaker of water: each swing carries the pointer to a slightly lower height, and a plot of amplitude versus time decays in a clean exponential envelope.

An external periodic driver can keep an oscillator going against damping. Push a child on a swing once each cycle, at just the right moment, and the swing maintains its amplitude. This is a forced oscillation: the body oscillates not at its own natural frequency but at the driver's frequency. The most striking case is resonance — the special configuration where the driver's frequency matches the system's natural frequency. The driver and driven reinforce each other; energy transfers maximally; amplitude grows to its largest value.

Resonance is more than a textbook curiosity. The collapse of the Tacoma Narrows suspension bridge in November 1940 — six months after it opened — happened because the wind, gusting at the bridge's natural frequency, drove its oscillations to amplitudes the structure could not bear. Marching soldiers were historically ordered to break step on bridges for the same reason. On a happier note, a radio tuner is an electrical resonator: turning the dial changes the natural frequency of its LC circuit until it matches the carrier frequency of the station you want, at which point that station's signal is amplified preferentially. A singer shattering a wine glass does so by sustaining a note at the glass's natural frequency — feeding energy in faster than the glass can dissipate it, until the strain exceeds the material's limit.

For NEET, the key cluster to remember is: free vibration (system left alone, oscillates at its natural frequency, no external force), damped vibration (resistive force present, amplitude decays, no driver), forced vibration (external driver, body oscillates at driver frequency), and resonance (the special forced case where driver frequency equals natural frequency, producing peak amplitude). These four ideas frame every advanced question on the topic, from atomic spectra to bridge engineering.