In short

Periodic motion repeats after equal time intervals. Oscillatory motion is periodic motion about an equilibrium position. Simple harmonic motion (SHM) is the special — and enormously common — case of oscillatory motion where the restoring force is proportional to the displacement and points back toward equilibrium:

F = -k x.

By Newton's second law this gives the equation

\frac{d^2 x}{dt^2} = -\omega^2 x, \quad \omega = \sqrt{k/m},

whose solution is x(t) = A\cos(\omega t + \phi) — a sinusoidal back-and-forth with amplitude A, angular frequency \omega, and phase \phi. The time period is T = 2\pi/\omega and the frequency is f = 1/T. Any system displaced slightly from a stable equilibrium oscillates at the frequency set by the curvature of its potential — which is why SHM turns up in a spring, a pendulum (for small angles), a floating plank, a swinging ceiling fan's pull chain, an LC circuit, a tuning fork, a vibrating guitar string, the atoms of a Diwali rangoli held together in a crystal lattice, and every other oscillation in physics that isn't violently far from equilibrium.

Hang a block from a spring, give it a gentle tug downward, and let go. The block bobs up and down with a rhythm so steady that you could set a metronome by it. Tap a tanpura string in a Carnatic concert and it quivers back and forth at the same rhythm for nearly a minute. Pull a grandfather clock's pendulum to one side and release — the swing takes precisely one second each way, century after century.

All three systems are doing the same thing. They have an equilibrium position (where the block rests at rest, where the string is straight, where the pendulum hangs straight down). When you displace them, something pulls or pushes them back toward that equilibrium. They overshoot, reverse, come back, overshoot the other way, and repeat. The motion has a specific mathematical signature — a sine wave in time — and a specific name: simple harmonic motion, or SHM.

SHM is the most important kind of oscillation in physics, not because it is complicated but because it is everywhere. Any stable system, when displaced slightly from equilibrium, oscillates with the motion of a SHM. The planet's orbits, the atoms in a crystal, the electrons in an LC circuit, the bob of a Kashi Vishwanath temple bell, the pressure of a sound wave in your ear, the current in the AM radio in your grandfather's drawing room — all of these are, at heart, simple harmonic motion. Master SHM once, and you have mastered the small-oscillation behaviour of every stable system in the universe.

This article — the first in the oscillations sequence — builds the concept from scratch. You will see what periodic and oscillatory motion mean, what makes SHM the clean special case, derive the equation of motion and its sinusoidal solution, meet the archetypal examples (spring-mass, small-angle pendulum, LC circuit, floating block), and understand why SHM is the universal behaviour near equilibrium. The next chapter — Equation of SHM and Phase — will open up the full vocabulary of amplitude, phase, velocity, acceleration, and energy.

Periodic motion and oscillatory motion — getting the vocabulary right

Start with two words that are close but not the same.

Periodic motion is any motion that repeats itself at regular intervals of time. The smallest time after which the motion looks identical to its earlier state is the time period T.

Oscillatory motion is a special kind of periodic motion: motion that goes back and forth about an equilibrium position. Not all periodic motion is oscillatory — a point on the rim of a rotating ceiling fan blade is periodic, but it does not reverse direction, so it is not oscillatory.

So: all oscillatory motion is periodic, but not all periodic motion is oscillatory. The defining feature of oscillation is the reversal — the object overshoots, comes back, overshoots again, and so on.

Three motion types: circular, linear, and oscillatory Three panels side by side. Left: a circle showing uniform circular motion — periodic but not oscillatory. Middle: a dashed horizontal line showing uniform motion — neither periodic nor oscillatory. Right: a horizontal segment with a block that moves back and forth between two extreme positions — the prototype of oscillatory motion. periodic (circular) not oscillatory neither periodic nor oscillatory oscillatory (back and forth about equilibrium) equilibrium
Three motion types. A body in uniform circular motion is periodic but never reverses, so it is not oscillatory. A body in uniform straight-line motion is neither periodic nor oscillatory. A body moving back and forth about an equilibrium position is oscillatory — the pattern we care about in this article.

The time period T is the time for one complete oscillation — one full back-and-forth. The frequency f = 1/T is the number of oscillations per second, measured in hertz (Hz). A tuning fork vibrating at 440 Hz (musical A) completes 440 full oscillations every second, with a time period of T = 1/440 \approx 2.3 ms.

What makes SHM special — the restoring force proportional to displacement

Every oscillation needs two ingredients: a place to come back to (the equilibrium), and a force that pulls the object back when it's displaced from that place. The pulling-back force is called the restoring force. In different oscillators, the restoring force takes different forms:

In general, the restoring force F_\text{rest}(x) is some function of the displacement x from equilibrium. At x = 0, the force is zero (that's the equilibrium). For small x, the force points back toward equilibrium — toward the origin — and gets larger the farther you go.

Simple harmonic motion is the clean special case where the restoring force is linear in the displacement:

\boxed{\; F_\text{rest}(x) = -k x \;}

with k > 0 a constant. The minus sign says "the force opposes the displacement": if you are to the right of equilibrium (x > 0), the force pushes you left. If you are to the left (x < 0), the force pushes you right. The bigger your displacement, the bigger the force, proportionally.

Force versus displacement for SHM — a straight line through origin with negative slope A coordinate plane with displacement x on the horizontal axis and restoring force F on the vertical axis. A straight red line passes through the origin with negative slope: F equals minus k x. Dashed vertical lines at x equals plus A and x equals minus A show the force at the extremes: minus k A at plus A, and plus k A at minus A. The arrows point back toward the origin in both cases. $x$ $F$ $O$ $F = -k x$ $+A$ $-kA$ $-A$ $+kA$
The defining signature of simple harmonic motion: the restoring force $F$ is a straight-line function of the displacement $x$, passing through the origin with a negative slope $-k$. At $x = +A$ (rightmost extreme), the force is $-kA$ (pushing left). At $x = -A$ (leftmost extreme), the force is $+kA$ (pushing right). The force always points back toward equilibrium.

The constant k is the force constant (also called the stiffness or, for a spring, the spring constant) — it has units of N/m. A stiffer system has a larger k, meaning a larger restoring force for a given displacement.

Why this is not as restrictive as it looks

"Restoring force proportional to displacement" might sound like a very specific condition — surely most systems don't exactly obey Hooke's law. True. But here is the remarkable fact: near any stable equilibrium, every smooth restoring force looks linear. Take any nice restoring function F(x) with F(0) = 0. Its Taylor expansion near x = 0 is

F(x) = F(0) + F'(0) x + \tfrac{1}{2} F''(0) x^2 + \cdots = -k x + O(x^2),

where -k \equiv F'(0) is the slope of F at the equilibrium (negative because the force pulls back). For small enough displacements, the higher-order terms are negligible, and the motion is SHM. This is why every stable system oscillates with SHM when you nudge it gently.

A pendulum, strictly, has restoring torque -mgL \sin\theta. For small angles, \sin\theta \approx \theta, and the torque becomes -mgL \theta — linear in \theta. Big-angle pendulums are anharmonic; small-angle pendulums are SHM. This "small-angle approximation" is the window in which the pendulum is a simple harmonic oscillator. Outside the window, the motion is still periodic but no longer simple-harmonic.

The equation of motion — and the sinusoidal solution

Take a spring-mass oscillator: a mass m attached to a spring of stiffness k, free to slide on a frictionless horizontal track. The natural length is at x = 0. Displace the mass to some position x and release.

By Newton's second law,

m \frac{d^2 x}{dt^2} = F = -k x.

Divide through by m:

\frac{d^2 x}{dt^2} = -\frac{k}{m} x \equiv -\omega^2 x,

where we have defined \omega = \sqrt{k/m}. This is the equation of motion for a simple harmonic oscillator.

Why: Newton's second law is F = ma. Acceleration is the second derivative of position, a = d^2 x/dt^2. Substituting F = -kx and dividing by m isolates the acceleration on the left. The quantity k/m has units of (\text{N/m})/\text{kg} = \text{s}^{-2}, which is \omega^2 — so \omega has units of \text{s}^{-1}, the correct unit for angular frequency. This substitution \omega^2 \equiv k/m is not a trick; it is the only combination of k and m with units of inverse time.

What function x(t) has the property that its second derivative is a negative constant times itself? The sine and cosine functions. Directly:

\frac{d^2}{dt^2}\cos(\omega t) = -\omega^2 \cos(\omega t),
\frac{d^2}{dt^2}\sin(\omega t) = -\omega^2 \sin(\omega t).

Either one solves the equation. The general solution is a linear combination:

x(t) = A\cos(\omega t) + B\sin(\omega t).

Trigonometry lets you rewrite this in the compact amplitude-phase form:

\boxed{\; x(t) = A_0 \cos(\omega t + \phi) \;}

where A_0 = \sqrt{A^2 + B^2} and \phi = -\arctan(B/A). The constants A_0 and \phi are set by initial conditions — by where the mass starts and how fast it is moving.

Why: the differential equation is linear, so any sum of solutions is also a solution. Both \sin(\omega t) and \cos(\omega t) satisfy it, so all their linear combinations do too. Writing the solution in amplitude-phase form consolidates the two constants A and B into one amplitude A_0 (how big the oscillation is) and one phase \phi (where in the cycle the oscillator starts at t = 0). The next chapter will explore both in detail.

The quantity \omega is the angular frequency (units: rad/s). The oscillation's time period is

T = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{m}{k}},

and the frequency is

f = \frac{1}{T} = \frac{1}{2\pi}\sqrt{\frac{k}{m}}.

Why: \cos repeats every 2\pi radians, so \omega t must advance by 2\pi for one full cycle, giving T = 2\pi/\omega. Frequency is 1 over period — the number of cycles per second. Stiffer system (bigger k) means higher frequency; heavier mass (bigger m) means lower frequency. Both are the intuitively correct directions: push a tight spring, it rebounds fast; push a massive object, it rebounds slow.

Watch SHM move — an animated spring-mass system

The animation below shows a spring-mass oscillator with m = 0.5 kg, k = 20 N/m. Its angular frequency is \omega = \sqrt{20/0.5} = \sqrt{40} \approx 6.32 rad/s, giving a time period of T = 2\pi/\omega \approx 0.99 s. The mass starts at x = 0.2 m to the right of equilibrium and is released from rest, so the motion is x(t) = 0.2 \cos(6.32\, t).

Animated: a mass on a spring executing simple harmonic motion A block attached to a spring on a horizontal surface. The block starts at x equals 0.2 metres to the right of equilibrium and is released from rest. It executes simple harmonic motion with amplitude 0.2 metres and time period approximately 1 second. Over the 6-second animation, the block completes about 6 full oscillations. equilibrium $x = 0$ $-A$ $+A$
A 0.5 kg mass attached to a 20 N/m spring. Released from $x = 0.2$ m with zero velocity, it oscillates with amplitude 0.2 m and time period $T \approx 1$ s. Watch the velocity — fastest at the centre (equilibrium), zero at the extremes. Ghost markers at $t = 0.25$ s, $0.5$ s, $0.75$ s, $1.0$ s show a quarter-period, half-period, three-quarter-period, and full-period positions: $+A$, $0$, $-A$, $0$, $+A$. Click replay.

Notice the shape of the motion. The block moves fastest when it passes through the equilibrium position (at x = 0), and it stops momentarily at the turning points (x = \pm A) before reversing. This is a direct consequence of the sine/cosine structure: x(t) = A\cos(\omega t), so v(t) = dx/dt = -A\omega \sin(\omega t). The speed is A\omega at x = 0 (where \sin = \pm 1) and 0 at x = \pm A (where \sin = 0).

Explore how period depends on mass and stiffness

Drag either parameter below to see how the SHM period responds. The formula is T = 2\pi\sqrt{m/k} — period grows as the square root of mass and shrinks as the square root of stiffness.

Interactive: SHM period versus mass for a fixed spring A curve showing the time period of simple harmonic motion as a function of mass for a spring of stiffness k equals 20 N per metre. The curve follows T equals 2 pi times root of m over k, starting at T equals 0 at m equals 0 and reaching approximately 3.14 seconds at m equals 5 kg. mass $m$ (kg) period $T$ (s) 0 1 2 3 4 1 2 3 4 5 drag the red point along the axis
Period $T = 2\pi\sqrt{m/k}$ as a function of mass (with $k = 20$ N/m). At $m = 0.5$ kg, $T \approx 0.99$ s (matching the animation above). Doubling the mass increases the period by a factor of $\sqrt{2} \approx 1.41$, not by 2 — the square-root dependence is why very heavy masses oscillate only slowly.

Archetypal SHM systems

The same equation — \ddot x = -\omega^2 x — describes a whole zoo of oscillating systems, each with its own angular frequency \omega set by the physical parameters. Here are four that every physics student should know by name.

1. Spring-mass oscillator

Already introduced. A mass m on a spring of stiffness k gives \omega = \sqrt{k/m} and T = 2\pi\sqrt{m/k}.

A ₹50 mobile-phone case with a spring-loaded pop-out stand, a clipboard clip, a ball-point pen spring — these are all spring-mass systems waiting to be released.

2. Simple pendulum (small angles)

A bob of mass m on a string of length L hangs straight down. Pull it aside by a small angle \theta and release. The restoring torque is -mgL\sin\theta, which for small \theta becomes -mgL\theta. Newton's second law for rotation (\tau = I\alpha, with I = mL^2) gives

mL^2 \frac{d^2\theta}{dt^2} = -mgL\theta \quad\Longrightarrow\quad \frac{d^2\theta}{dt^2} = -\frac{g}{L}\theta.

This is SHM in \theta with \omega^2 = g/L, so

T = 2\pi\sqrt{L/g}.

Remarkably, the mass does not appear in the period. A heavy bob and a light bob on strings of the same length swing at exactly the same rate. Galileo noticed this watching a chandelier swing in a Pisa cathedral; the full derivation is in Simple Pendulum.

3. LC circuit (electrical SHM)

An inductor L and capacitor C in a loop oscillate exactly like a spring-mass system — but with charge and current playing the roles of position and velocity. The equation is

L\frac{d^2 Q}{dt^2} = -\frac{Q}{C} \quad\Longrightarrow\quad \frac{d^2 Q}{dt^2} = -\omega^2 Q,

with \omega = 1/\sqrt{LC}. The charge on the capacitor oscillates sinusoidally, and so does the current. This is the mathematical heart of every radio tuner, every electronic clock, every FM transmitter — all resonant circuits exchanging energy between the electric field of the capacitor and the magnetic field of the inductor, with \omega^2 = 1/(LC) setting the resonant frequency.

4. Floating block (buoyancy as the restoring force)

A rectangular block of uniform density floats partly submerged in water. Push it down by a small distance y; by Archimedes, extra buoyancy pushes it back up by F = -\rho_w g A y (where A is the cross-sectional area). If the block has mass m, Newton's second law gives

m \frac{d^2 y}{dt^2} = -\rho_w g A y,

which is SHM with \omega = \sqrt{\rho_w g A/m}. A cork bob on the surface of a pond, the kalash floated on the Ganges during a puja, a tennis ball held under water and released — all of these are SHM in disguise.

Why these all look the same

In each case the mathematical skeleton is identical: (inertial quantity) × (second time derivative of a variable) = −(elastic quantity) × (that variable). The inertia can be mass, moment of inertia, inductance. The elastic quantity can be a spring constant, a gravitational pendulum constant, the inverse of capacitance, or the buoyancy stiffness. Once you see the pattern, you can spot SHM in any new system you meet and write down its frequency almost immediately.

Worked examples

Example 1: Spring-mass — compute frequency and period

A toy block of mass m = 200 g is attached to a horizontal spring of stiffness k = 50 N/m on a frictionless surface. The block is pulled 4 cm to the right of equilibrium and released from rest. Find (a) the angular frequency, (b) the time period, (c) the frequency, and (d) the maximum speed of the block.

A block on a spring on a horizontal surface A horizontal surface with a wall at the left end. A spring extends horizontally from the wall to a rectangular block. The block is shown displaced 4 cm to the right of its equilibrium position, labelled x equals 0. $m$ $x = 0$ $x = 4$ cm
A block on a spring is pulled 4 cm to the right of equilibrium and released from rest. The motion that follows is simple harmonic.

Step 1. Angular frequency.

\omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{50}{0.200}} = \sqrt{250} \approx 15.81 \text{ rad/s}.

Why: convert mass to SI (200 g = 0.200 kg) before dividing. Square-rooting the ratio of stiffness to mass gives the angular frequency directly — a single-step computation.

Step 2. Time period.

T = \frac{2\pi}{\omega} = \frac{2\pi}{15.81} \approx 0.397 \text{ s}.

Step 3. Frequency.

f = \frac{1}{T} = \frac{1}{0.397} \approx 2.52 \text{ Hz}.

Why: frequency is the reciprocal of period — the number of oscillations per second. Here the block completes about 2.5 full back-and-forth cycles in every second.

Step 4. Maximum speed.

The amplitude is A = 0.04 m (the initial displacement, since released from rest). The motion is x(t) = 0.04\cos(15.81 \, t), so

v(t) = \frac{dx}{dt} = -0.04 \times 15.81 \sin(15.81\, t).

The maximum value of |\sin| is 1, so v_\text{max} = 0.04 \times 15.81 \approx 0.632 \text{ m/s}.

Why: max speed is amplitude times angular frequency — v_\text{max} = A\omega. This occurs as the block passes through the equilibrium position, where all the spring's stored potential energy has been converted to kinetic. The next chapter on energy in SHM derives this from energy conservation.

Result: \omega \approx 15.8 rad/s; T \approx 0.40 s; f \approx 2.5 Hz; v_\text{max} \approx 0.63 m/s.

What this shows: A light mass on a moderate spring oscillates several times a second. You can hear this — spring-loaded school-bag clips, the tak-tak of a door spring, the pure ring of a tuning fork all live in the tens-to-hundreds-of-hertz range set by their \sqrt{k/m}.

Example 2: Pendulum — the length of a 1 second pendulum

A simple pendulum has a time period of T = 2.00 s — it swings one way in 1 second, the other way in 1 second, so it ticks each second. (This is called a "seconds pendulum," the standard used in pre-quartz grandfather clocks.) Find the length L of the pendulum, assuming g = 9.80 m/s² at sea level in Mumbai.

A simple pendulum with length L and angular displacement theta A small circle representing the pendulum support at the top. A straight line of length L descends at an angle theta to the vertical, ending in a bob (a larger circle). A vertical dashed line from the support shows the equilibrium position. The angle theta is marked with an arc between the string and the vertical. $\theta$ $L$ bob: mass $m$ equilibrium
A simple pendulum. The length $L$ is the distance from the pivot to the bob. For small $\theta$, the motion is SHM with $T = 2\pi\sqrt{L/g}$.

Step 1. Write the pendulum period formula.

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

Step 2. Solve for L.

Square both sides:

T^2 = 4\pi^2 \frac{L}{g} \quad\Longrightarrow\quad L = \frac{T^2 g}{4\pi^2}.

Why: isolate L by squaring to remove the square root, then rearrange. The 4\pi^2 in the denominator is what you might forget — it comes from squaring 2\pi.

Step 3. Substitute numbers.

L = \frac{(2.00)^2 \times 9.80}{4 \pi^2} = \frac{4 \times 9.80}{39.48} \approx 0.993 \text{ m}.

Why: 4\pi^2 \approx 39.48. The answer comes out to just under 1 metre, which is a useful fact — a seconds pendulum is almost exactly 1 m long at sea level. This is not a coincidence of nature; early definitions of the metre were actually proposed in terms of the length of a seconds pendulum.

Result: L \approx 0.99 m.

What this shows: A grandfather clock's pendulum is about 1 metre long — the natural length of anything that ticks each second. Shorter pendulums swing faster (a kitchen spice jar on a string swings several times per second); longer ones swing slower (the Foucault pendulum at the Birla Science Museum in Hyderabad, about 25 m long, has T \approx 10 s).

Example 3: LC circuit — find the resonant frequency

A tuning circuit in a portable AM radio has an inductor L = 250 μH and a variable capacitor currently set to C = 100 pF. Find the resonant angular frequency, the resonant frequency, and identify which AM broadcast band this falls in (if any).

An LC circuit diagram A rectangular circuit loop. An inductor (standard coil symbol) is on the left side. A capacitor (two parallel plates) is on the right side. The top and bottom horizontal wires connect them. Labels L and C mark the components. $L = 250$ μH $C = 100$ pF
An LC circuit with inductance $L$ and capacitance $C$. Charge oscillates back and forth between the capacitor plates at the resonant frequency $\omega = 1/\sqrt{LC}$.

Step 1. Write the resonance condition.

\omega = \frac{1}{\sqrt{LC}}.

Step 2. Convert to SI units. L = 250 \times 10^{-6} H; C = 100 \times 10^{-12} F.

LC = 250 \times 10^{-6} \times 100 \times 10^{-12} = 2.5 \times 10^{-14} \text{ s}^2.
\sqrt{LC} = \sqrt{2.5 \times 10^{-14}} \approx 1.58 \times 10^{-7} \text{ s}.

Step 3. Compute \omega and f.

\omega = \frac{1}{1.58 \times 10^{-7}} \approx 6.32 \times 10^6 \text{ rad/s}.
f = \frac{\omega}{2\pi} = \frac{6.32 \times 10^6}{6.28} \approx 1.01 \times 10^6 \text{ Hz} = 1.01 \text{ MHz}.

Why: same formula as the spring-mass system with the substitutions k \to 1/C and m \to L. That the electrical oscillator frequency follows the same pattern as the mechanical one is not a coincidence — the underlying differential equation is identical.

Result: The circuit resonates at approximately 1.01 MHz.

What this shows: The Indian AM broadcast band runs from 535 kHz to 1605 kHz. 1.01 MHz = 1010 kHz lies squarely in the middle — in the evening you might tune this circuit to AIR Vividh Bharati at 1134 kHz or to Radio Mirchi's music service at 980 kHz by slightly adjusting the variable capacitor. Rotating the capacitor knob changes C, which changes \omega by 1/\sqrt{LC}, which changes the station the radio picks out. Every AM radio in the country is, at its heart, an LC oscillator.

Common confusions

If you have the definition of SHM, the equation of motion, and can identify and compute the period of standard oscillators, you have the working content of this article. What follows is the connection between SHM and uniform circular motion (the original definition and a beautiful geometric picture), the universality argument via Taylor expansion, and the first glimpse of the energy picture that will be developed fully in Energy in SHM.

SHM as the shadow of uniform circular motion

Take a particle moving in a circle of radius A at constant angular speed \omega. Its x-coordinate is

x(t) = A\cos(\omega t + \phi).

This is exactly the equation of a simple harmonic oscillator. So the x-component of uniform circular motion is simple harmonic motion. The two motions are mathematically the same projection of one onto the other.

SHM as the projection of uniform circular motion onto a diameter A circle of radius A centred at origin, with a point P on the circle at angle omega t plus phi from the positive x axis. A vertical dashed line from P drops to the x axis at a foot labelled P prime at position x equals A cosine of omega t plus phi. As P moves uniformly around the circle, P prime oscillates back and forth along the x axis with simple harmonic motion. $x$ $y$ $O$ $P$ $P'$ $x$ $\omega t$ SHM on $x$-axis (shadow of $P$)
SHM as the shadow of uniform circular motion. A point $P$ moves at constant angular speed $\omega$ around a circle of radius $A$. Its shadow $P'$ on the $x$-axis moves back and forth as $x(t) = A\cos(\omega t + \phi)$ — simple harmonic motion.

This is the historical origin of the word simple harmonic: the Greek notion of harmony was based on circular motion, and the fact that the "shadow" of perfectly regular circular motion is sinusoidal gives the motion its "harmonic" character. It is also the origin of the name \omega — it is the angular speed of the corresponding circular motion, even in systems where there is no actual circle (spring-mass, LC circuit). The SHM shadow picture is why calling \omega the "angular frequency" and using the unit rad/s makes sense — it is a leftover from the circular-motion interpretation.

The shadow picture is also a concrete way to see why v_\text{max} = A\omega and a_\text{max} = A\omega^2: v_\text{max} is the speed of P around the circle projected onto the x-axis, and a_\text{max} is the centripetal acceleration of P projected similarly.

The universal argument — why SHM is everywhere

Consider a particle with potential energy U(x) sitting at a stable equilibrium at x = 0. "Stable" means U has a local minimum there: U'(0) = 0 (no force at equilibrium) and U''(0) > 0 (curves upward — any small displacement costs energy, so there is a restoring tendency).

Expand U(x) as a Taylor series around x = 0:

U(x) = U(0) + U'(0) x + \tfrac{1}{2} U''(0) x^2 + \tfrac{1}{6} U'''(0) x^3 + \cdots

Set U(0) = 0 (choose the potential's zero at equilibrium) and U'(0) = 0 (equilibrium condition). The leading non-trivial term is \tfrac{1}{2} U''(0) x^2. For small x the higher-order terms are negligible compared to this quadratic term, so

U(x) \approx \tfrac{1}{2} k x^2, \quad k \equiv U''(0) > 0.

This is the harmonic potential. The restoring force is

F(x) = -\frac{dU}{dx} = -k x,

which is Hooke's law. So: every stable system, for small displacements, has a harmonic potential and obeys Hooke's law to leading order. This is the universality of SHM — why the same mathematics describes spring oscillators, pendulums, crystals, and radio-tuning circuits. The physical identity of the oscillator doesn't matter; only the second derivative of the energy at equilibrium does.

This argument fails only when U''(0) = 0 — when the potential is flat at equilibrium to quadratic order. In such cases the oscillation is not SHM but something with a different power-law period dependence. The two-dimensional Mexican-hat potential, the rotation of a rigid body about an axis of intermediate moment of inertia, and a ball rolling in a perfectly flat-bottomed bowl are all examples. But the "generic" case is SHM, and you will meet the exceptions as exceptions.

First glimpse of energy in SHM

The kinetic energy of a SHM oscillator is \tfrac{1}{2}m v^2 = \tfrac{1}{2} m A^2 \omega^2 \sin^2(\omega t + \phi), and the potential energy is \tfrac{1}{2} k x^2 = \tfrac{1}{2} k A^2 \cos^2(\omega t + \phi). Using k = m\omega^2,

E = \tfrac{1}{2} m A^2 \omega^2 [\sin^2(\omega t + \phi) + \cos^2(\omega t + \phi)] = \tfrac{1}{2} m A^2 \omega^2 = \tfrac{1}{2} k A^2.

The total energy is constant, and it depends only on the amplitude and the stiffness, not on the phase. The KE and PE swap back and forth but their sum is fixed. This is conservation of mechanical energy for the oscillator — fully developed in Energy in SHM. The ratio E/A^2 = (constant depending on system) is often a cleaner parameter for comparing oscillators than the amplitude alone.

Damping and driving — beyond SHM

Real oscillators do not oscillate forever — friction and air resistance drain energy, slowly shrinking the amplitude. This is damped oscillation, governed by

m\ddot x + b\dot x + k x = 0.

If an oscillator is driven by an external periodic force F_0 \cos(\Omega t), the response is forced oscillation:

m\ddot x + b\dot x + k x = F_0 \cos(\Omega t).

Both extensions produce rich physics — damped decay, beats, resonance catastrophe, the specific shape of the response-vs-frequency curve — all built on top of the basic SHM framework developed here. The next few chapters will take these up in turn.

An Indian footnote

The jantar-mantar observatories of Jaipur and Delhi (built in the 1720s by Sawai Jai Singh II) contain the Samrat Yantra — the world's largest sundial, whose gnomon's shadow sweeps across the Earth once per day. As the shadow moves along a calibrated arc, its rate corresponds to the angular speed of the Sun's apparent motion — which is essentially uniform circular motion (geocentrically) — and the shadow's motion along any one axis is the SHM shadow picture described above, on a scale of hours rather than seconds. The design of the sundial anticipates, in physical form, the connection between rotation and harmonic oscillation that Hooke's and Newton's mathematics would formalise a generation later.

Where this leads next