Periodic vs oscillatory motion
Among the motions you have studied, rectilinear motion and projectile motion are non-repetitive. Uniform circular motion and the orbital motion of planets are repetitive: they return to the same configuration after a fixed interval. Any motion that repeats itself at regular intervals of time is called periodic motion. Rocking in a cradle, a swinging pendulum, a boat tossing up and down, the piston in a steam engine — these repeat too, but with an extra feature. The body moves to and fro about a fixed central position. Such to-and-fro periodic motion about a mean position is called oscillatory motion.
The two classes are nested, not equal. NCERT states the relationship in one line: every oscillatory motion is periodic, but every periodic motion need not be oscillatory. Oscillation is the special case of periodic motion that carries a mean position and a swinging back and forth. Vibration is the same idea under a different name — there is no significant physical difference, only a loose convention that low-frequency repetition is called oscillation (a tree branch swaying) and high-frequency repetition is called vibration (a guitar string).
Periodic motion
- Repeats itself at regular intervals of time.
- Needs a fixed period \(T\); needs no mean position.
- May or may not reverse direction.
- Examples: planet around the Sun, hands of a clock, uniform circular motion, a bouncing ball.
- All oscillatory motions live inside this set.
Oscillatory (vibratory) motion
- Periodic and to and fro about a fixed equilibrium.
- A restoring force pulls the body back to the mean position.
- Always reverses direction each half cycle.
- Examples: simple pendulum, mass on a spring, string of a sitar.
- Every such motion is automatically periodic.
Equilibrium and the restoring force
What singles out an oscillation from a generic periodic motion is an equilibrium position lying somewhere inside the path. At this position no net external force acts on the body, so if it is left there at rest it stays there forever. Displace it a little and a force comes into play that tries to drag it back toward equilibrium. That restoring tendency is precisely what produces the to-and-fro swing. A ball resting at the bottom of a bowl is the standard picture: nudge it sideways and it rolls back and overshoots, oscillating about the lowest point.
The simplest oscillation arises when this restoring force is directly proportional to the displacement from the mean position and always directed toward it. That special case is simple harmonic motion, the backbone of the entire chapter. Real oscillators eventually stop because friction and other dissipative causes drain their energy — this is damping — and they can be kept going by an external periodic driver.
Examples that sort the two classes
The cleanest way to internalise the distinction is to classify motions by hand. The table below sorts familiar motions; the figure that follows contrasts a planetary orbit (periodic, not oscillatory) with a pendulum (both).
| Motion | Periodic? | Oscillatory? | Why |
|---|---|---|---|
| Planet orbiting the Sun | Yes | No | Repeats every revolution, but never reverses about a mean position |
| Hands of a clock | Yes | No | Uniform rotation; periodic but not to and fro |
| Uniform circular motion | Yes | No | Returns to the same point each period; no equilibrium inside the path |
| Simple pendulum (small swing) | Yes | Yes | To and fro about the vertical mean position |
| Mass on a spring | Yes | Yes | Oscillates about the natural-length equilibrium |
| String of a sitar / guitar | Yes | Yes (vibration) | High-frequency to-and-fro about the rest line |
| Projectile flight | No | No | Single non-repeating path |
Period and frequency
The smallest interval of time after which the motion repeats is called the period, denoted \(T\), with SI unit the second. For motions too fast or too slow for seconds, convenient sub- and super-units are used: a quartz crystal vibrates with a period of microseconds (\(10^{-6}\,\text{s}\)), while Mercury's orbital period is 88 earth days and Halley's comet returns every 76 years.
The reciprocal of the period gives the number of repetitions per unit time. This is the frequency, denoted \(\nu\) (Greek nu), related to the period by
$$\nu = \frac{1}{T}.$$
The unit of frequency is \(\text{s}^{-1}\), given the special name hertz (Hz) after Heinrich Hertz: \(1\ \text{Hz} = 1\ \text{oscillation per second} = 1\ \text{s}^{-1}\). The frequency need not be an integer.
On average a human heart beats 75 times in a minute. Calculate its frequency and period.
Frequency. \(\nu = \dfrac{75}{60\ \text{s}} = 1.25\ \text{s}^{-1} = 1.25\ \text{Hz}.\)
Period. \(T = \dfrac{1}{\nu} = \dfrac{1}{1.25\ \text{s}^{-1}} = 0.8\ \text{s}.\) The heart repeats its cycle once every 0.8 seconds.
Angular frequency ω
A third quantity completes the trio. The angular frequency \(\omega\) measures the rate at which the phase of the motion advances, in radians per second. It connects to both period and frequency:
$$\omega = \frac{2\pi}{T} = 2\pi\nu.$$
The factor \(2\pi\) is the number of radians in one complete cycle, so \(\omega\) is simply \(2\pi\) times the ordinary frequency. The three descriptors say the same thing in three currencies, and any one fixes the other two.
| Quantity | Symbol | Meaning | SI unit | Relations |
|---|---|---|---|---|
| Period | \(T\) | Smallest time for one full repeat | second (s) | T = 1/ν = 2π/ω |
| Frequency | \(\nu\) | Repeats per unit time | hertz (Hz) | ν = 1/T = ω/2π |
| Angular frequency | \(\omega\) | Rate of phase advance | rad s⁻¹ | ω = 2π/T = 2πν |
Angular frequency is the same \(\omega\) that turns a reference circle — see SHM and uniform circular motion for the geometric origin of \(\omega\).
Displacement as a function of time
In kinematics, displacement meant the change in a particle's position vector. In oscillations the term is generalised: it refers to the change with time of any physical property under consideration. The choice of origin is a matter of convenience, and it is usual to measure displacement from the equilibrium position.
The displacement variable therefore wears many costumes depending on the system, and it can take both positive and negative values about the mean. NEET items often hide this generality inside a phrase such as "displacement of an AC voltage" — fully legitimate under this definition.
| Oscillating system | Displacement variable |
|---|---|
| Block on a spring | Linear distance \(x\) from equilibrium |
| Simple pendulum | Angular displacement \(\theta\) from the vertical |
| AC circuit | Voltage across a capacitor |
| Sound wave | Pressure variation in the medium |
| Light wave | Changing electric and magnetic fields |
Because the motion repeats, the displacement is a periodic function of time, \(f(t) = f(t+T)\). The waveform below shows a generic periodic displacement: the same shape recurs after each period \(T\), and the maximum excursion on either side of the mean is the amplitude.
Periodic functions and the Fourier idea
The displacement can be written as a mathematical function of time, and for periodic motion that function is periodic. The simplest such functions are the cosine and sine:
$$f(t) = A\cos\omega t, \qquad f(t) = A\sin\omega t.$$
Increasing the argument \(\omega t\) by any integer multiple of \(2\pi\) radians leaves the value unchanged, so each is periodic with period \(T = 2\pi/\omega\), confirming \(f(t) = f(t+T)\). A linear combination keeps the same period:
$$f(t) = A\sin\omega t + B\cos\omega t = D\sin(\omega t + \phi),$$
where \(D = \sqrt{A^2 + B^2}\) and \(\tan\phi = B/A\). The combination is still a single sinusoid of period \(T\), merely shifted in phase. This is the algebraic root of the standard SHM form \(x = D\sin(\omega t + \phi)\).
The deeper statement, proved by Jean Baptiste Joseph Fourier, lifts sine and cosine to a privileged status: any periodic function can be expressed as a superposition of sine and cosine functions of different periods with suitable coefficients. However jagged a periodic signal looks, it is built from simple sinusoids. This is why the chapter studies simple harmonic motion first — the sinusoidal oscillation is the elementary brick from which every periodic motion is assembled.
Which of the following functions of time represent periodic motion, and what is the period (\(\omega\) is a positive constant)? (i) \(\sin\omega t + \cos\omega t\) (ii) \(\sin\omega t + \cos 2\omega t + \sin 4\omega t\) (iii) \(e^{-\omega t}\)
(i) Both terms have period \(2\pi/\omega\); their sum is the single sinusoid \(\sqrt{2}\sin(\omega t + \pi/4)\). Periodic, with period \(T = 2\pi/\omega\).
(ii) The terms have periods \(2\pi/\omega,\ \pi/\omega,\ \pi/2\omega\). The smallest common multiple is \(2\pi/\omega\), so the sum is periodic with \(T = 2\pi/\omega\).
(iii) \(e^{-\omega t}\) decreases monotonically toward zero and never repeats. It is non-periodic.
Periodic & oscillatory motion in one breath
- Periodic motion repeats at regular intervals; oscillatory motion is periodic and to and fro about a fixed equilibrium.
- Every oscillatory motion is periodic; the converse is false — a planetary orbit is periodic but not oscillatory.
- An oscillation has an equilibrium position and a restoring force directed back toward it.
- Period \(T\) (seconds) and frequency \(\nu\) (hertz) are reciprocals: \(\nu = 1/T\).
- Angular frequency \(\omega = 2\pi/T = 2\pi\nu\) — the \(2\pi\) is essential, \(\omega \neq \nu\).
- Displacement is generalised to any physical quantity (position, angle, voltage, pressure) measured from the mean.
- Sine and cosine are the simplest periodic functions; by Fourier, every periodic function is a sum of sines and cosines.