Oscillations

1. What this chapter covers

A swing, a plucked guitar string, an AC voltage, and a vibrating atom in a solid all share the same underlying mathematics — this chapter builds that mathematics once, from a single restoring-force idea, and then applies it everywhere.

2. Periodic motion is not the same thing as oscillatory motion

A motion that repeats itself at regular time intervals is periodic. The smallest such interval is the period $T$; its reciprocal is the frequency, $\nu =1/T$, measured in hertz (1 Hz = 1 oscillation per second).

Every oscillatory motion — repeated back-and-forth movement about a mean position — is periodic. But the reverse is not true: a planet orbiting the Sun, or the Earth spinning on its axis, is periodic without ever oscillating back and forth. The chapter's opening line draws this boundary precisely because exam questions live exactly on it: "every oscillatory motion is periodic, but every periodic motion need not be oscillatory."

Displacement in this chapter means something broader than position. It is whatever quantity varies with time in the periodic motion under study — position for a spring-mass system, angle from vertical for a pendulum, even voltage across a capacitor in an AC circuit or pressure in a sound wave.

Worked example. A human heart beats 75 times per minute on average. Frequency $\nu =75/60\text{s}=1.25$ Hz. Period $T=1/\nu =1/1.25=0.8$ s — a concrete reminder that period and frequency describe the same repetition, just measured two different ways.

3. Simple harmonic motion: one specific kind of periodic function

Not every periodic displacement qualifies as simple harmonic motion (SHM). SHM is the specific case where displacement varies sinusoidally with time:

$x(t)=A\cos (\omega t+\varphi )$

• $A$ — amplitude, the maximum displacement from the mean position.
• $\omega t+\varphi$ — the phase, a time-dependent quantity.
• $\varphi$ — the phase constant, the phase's value at $t=0$.
• $\omega$ — the angular frequency, related to the period by $\omega =2\pi /T$.

A useful test for whether a given function of time is SHM: it must be expressible as $A\cos \omega t+B\sin \omega t$ for constants $A$ and $B$ — a single sine or cosine term, or a combination of sine and cosine at the same frequency.

A sum of terms at different frequencies, like $\cos \omega t+\cos 3\omega t$, is still periodic (Fourier's theorem guarantees any periodic function can be built from enough sine and cosine terms), but it is not SHM.

Phase difference between two SHMs of the same frequency is simply the difference in their $\varphi$ values. Two motions in phase ($\Delta \varphi =0$) reach their extremes together; two motions $\pi$ out of phase reach opposite extremes at the same instant; two motions $\pi /2$ out of phase have one at its extreme exactly when the other is crossing its mean position.

This single idea — comparing phase constants, not comparing the motions point by point — is what makes superposition-of-SHM problems tractable.

4. The circle hiding inside every SHM

Here is the chapter's most quietly remarkable result. Tie a ball to a string and swing it in a horizontal circle at constant angular speed — genuine uniform circular motion. Now look at it edge-on, so you only see its motion along one diameter. What you see is indistinguishable from a mass oscillating on a spring.

Formally: if a particle P moves uniformly on a circle of radius $A$ with angular speed $\omega$, starting at angle $\varphi$ from the x-axis, its projection onto the x-axis has position

$x(t)=A\cos (\omega t+\varphi )$

which is exactly the SHM equation. The circle is called the reference circle, and P the reference particle. Projecting onto the y-axis instead gives $y(t)=A\sin (\omega t+\varphi )$ — an SHM of the same amplitude and frequency, just $\pi /2$ out of phase with the x-projection.

This connection is a calculation trick, not a claim about forces: the force needed to keep the real ball moving in a circle (centripetal, always perpendicular to velocity) is nothing like the force driving the linear oscillator (always along the line of motion, toward the mean position). Only the displacement pattern matches.

Worked example. A reference particle P moves anticlockwise on a circle of radius $A$, period $T=4$ s, starting at $t=0$ on the +x-axis ($\varphi =0$). Its x-projection is $x(t)=A\cos \left(\frac{2\pi }{4}t\right)=A\cos \left(\frac{\pi }{2}t\right)$ — an SHM of the same period as the circular motion, with amplitude equal to the circle's radius.

If instead P starts on the $-x$-axis ($\varphi =\pi$) moving clockwise, the projection becomes $x(t)=A\cos \left(-\frac{\pi }{2}t+\pi \right)=-A\cos \left(\frac{\pi }{2}t\right)$ — same amplitude and period, but shifted in phase by exactly $\pi$, since clockwise rotation is what reverses the sign inside the cosine.

5. Velocity and acceleration follow by differentiating twice

Differentiating $x(t)=A\cos (\omega t+\varphi )$ with respect to time gives velocity, and differentiating again gives acceleration:

$v(t)=-\omega A\sin (\omega t+\varphi ),a(t)=-{\omega }^{2}A\cos (\omega t+\varphi )=-{\omega }^{2}x(t)$

The result worth carrying forward is the last equality: acceleration is always proportional to displacement, and always points opposite to it — toward the mean position, regardless of which side of it the particle is on.

Displacement, velocity, and acceleration all oscillate with the same period $T$, but they are staggered in phase: velocity leads displacement by $\pi /2$, and acceleration is exactly $\pi$ out of phase with displacement (maximal at the extremes, zero at the mean position — precisely when displacement is doing the opposite).

Worked example. A body oscillates as $x(t)=5\cos (2\pi t+\pi /4)$ m. Its angular frequency is $\omega =2\pi {\text{s}}^{-1}$. At $t=1.5$ s, the phase is $2\pi (1.5)+\pi /4=3\pi +\pi /4$, which lands in the same place on the cycle as $\pi +\pi /4$.

Displacement: $x=5\cos (3\pi +\pi /4)=-3.54$ m. Velocity: $v=-\omega A\sin (3\pi +\pi /4)=-22$ m/s. Acceleration: $a=-{\omega }^{2}x=-(2\pi {)}^2(-3.54)=140{\text{m/s}}^2$ — note that $a$ and $x$ have opposite signs, exactly as $a=-{\omega }^{2}x$ requires.

6. The force law — and why it is the deeper definition of SHM

Combining $a(t)=-{\omega }^{2}x(t)$ with Newton's second law gives the force on a particle of mass $m$ undergoing SHM:

$F(t)=ma(t)=-m{\omega }^{2}x(t)=-kx(t),k=m{\omega }^2$

This is Hooke's law, and $k$ is the familiar spring (force) constant. A particle governed by this restoring force is a linear harmonic oscillator; the force always points back toward the mean position, which is why it is also called the restoring force.

SHM can be defined two completely equivalent ways: by its displacement equation (Section 3) or by this force law. Differentiating the displacement equation twice produces the force law; integrating the force law twice recovers the displacement equation. Whichever a problem hands you, the other follows.

Worked check. A block of mass $m$ sits between two identical springs of constant $k$, each fixed to a wall on either side. Displace the block by $x$: the left spring stretches by $x$, exerting a restoring force $-kx$; the right spring compresses by $x$, exerting a restoring force $-kx$ as well — the same sign, since a compressed spring pushes back the same way a stretched one pulls.

The net force is $F=-2kx$, still linear in $x$, so the motion is SHM with an effective spring constant $2k$: $T=2\pi \sqrt{m/2k}$.

7. Energy in SHM: kinetic and potential trade places, total stays fixed

Kinetic energy depends on speed, so it is zero at the extremes (where $v=0$) and maximum at the mean position:

$K=\frac{1}{2}m{v}^2=\frac{1}{2}m{\omega }^2{A}^2{\sin }^2(\omega t+\varphi )=\frac{1}{2}k{A}^2{\sin }^2(\omega t+\varphi )$

The restoring force $F=-kx$ is conservative, with associated potential energy $U(x)=\frac{1}{2}k{x}^2$ — zero at the mean position, maximum at the extremes, exactly the opposite pattern to $K$:

$U=\frac{1}{2}k{A}^2{\cos }^2(\omega t+\varphi )$

Adding them, the ${\sin }^2+{\cos }^2=1$ identity collapses the time-dependence entirely:

$E=K+U=\frac{1}{2}k{A}^2$

Total mechanical energy is constant — exactly what conservation of energy demands for motion under a conservative force, and a useful check on any SHM numerical: compute $E$ once, and $K$ at any displacement follows immediately as $E-\frac{1}{2}k{x}^2$, without needing velocity at all.

Worked example. A 1 kg block on a spring of constant $k=50\text{N/m}$ is pulled to $x=10$ cm and released from rest. Total energy: $E=\frac{1}{2}k{A}^2=\frac{1}{2}(50)(0.1{)}^2=0.25$ J, fixed for the whole motion. At $x=5$ cm: $U=\frac{1}{2}(50)(0.05{)}^2=0.0625$ J, so $K=E-U=0.1875$ J — found without ever computing a velocity.

8. The simple pendulum: SHM is an approximation, not exact

A simple pendulum — a bob of mass $m$ on a massless, inextensible string of length $L$ — is not exactly SHM. Its exact restoring torque about the support is $\tau =-L(mg\sin \theta )$, and $\sin \theta$ is not linear in $\theta$.

For small angles, though, $\sin \theta \approx \theta$ (the series $\sin \theta =\theta -{\theta }^3/3!+\dots$ loses almost nothing once the cubic term is negligible — true within about 1% up to roughly $20°$). Substituting this approximation:

$\alpha =-\frac{mgL}{I}\theta$

which has exactly the form $a=-{\omega }^{2}x$ from Section 5, with angular displacement $\theta$ standing in for $x$. This proves the pendulum's motion is SHM only for small oscillations — a genuinely approximate result, not an exact one, unlike the spring-mass system.

Since the string is massless, $I=m{L}^2$, and Eq. above gives $\omega =\sqrt{g/L}$, so:

$T=2\pi \sqrt{\frac{L}{g}}$

Notably, $m$ cancels out entirely — a pendulum's period depends on its length and local gravity alone, never on the mass of the bob.

9. A textbook loose end worth knowing about

The chapter's own introduction promises to "discuss the phenomena of damped and forced oscillations later in the chapter" — but the rationalised 2026-27 edition's section list stops at 13.8, The Simple Pendulum, and neither topic appears anywhere in the body. This is a leftover sentence from an older edition that did cover them.

CBSE's own Unit X syllabus line for this chapter does not list damped or forced oscillations either, so nothing here is actually missing from what gets examined — but if a source ever asks for those two words "explained by this chapter," they were quietly dropped from the rationalised syllabus, not overlooked here.

Summary

• Periodic motion repeats at regular intervals; oscillatory motion is periodic motion that goes back and forth about a mean position — every oscillation is periodic, not every periodic motion oscillates.
• SHM is periodic motion whose displacement is specifically sinusoidal, $x(t)=A\cos (\omega t+\varphi )$, not just any repeating pattern.
• The projection of uniform circular motion onto a diameter is SHM — a geometric coincidence useful for solving problems, not a statement about the forces involved.
• $v(t)=-\omega A\sin (\omega t+\varphi )$ and $a(t)=-{\omega }^{2}x(t)$ follow by differentiating the displacement equation twice; acceleration always opposes displacement.
• The force law $F=-kx$, with $k=m{\omega }^2$, is Hooke's law and an equivalent definition of SHM to the displacement equation.
• Total mechanical energy $E=\frac{1}{2}k{A}^2$ is constant; kinetic and potential energy trade off between zero and this maximum as the particle moves.
• A simple pendulum's motion is SHM only approximately, valid for small angles where $\sin \theta \approx \theta$; its period $T=2\pi \sqrt{L/g}$ is independent of the bob's mass.