Define a periodic function mathematically. Are all periodic functions examples of simple harmonic motion? Just

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Q. Define a periodic function mathematically. Are all periodic functions examples of simple harmonic motion? Justify your answer with an example of a motion that is periodic but is NOT S.H.M.

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Concept understanding — Frequency And Period

Frequency and Period: The Rhythm of Repetition

Imagine a child on a swing, going back and forth. Each full cycle — forward, backward, back to the start — takes a certain amount of time. That time is the period. Now count how many such cycles happen in one second — that count is the frequency. Frequency and period describe the same repeating rhythm, just viewed from two different angles.

Everyday Intuition

Think of your own heartbeat. At rest, your heart beats roughly once every second. The period is about 1 second — the time between one beat and the next. The frequency is about 1 beat per second — how many beats occur in that same second.

If your heart speeds up, the period shrinks (less time between beats) and the frequency rises (more beats per second). Slow the heart down, and it's the reverse.

Note

Period and frequency are inversely related — when one increases, the other decreases. Always.

Precise Definitions

Period ( $T$ ) — the time taken to complete one full cycle of a repeating event, measured in seconds (s).

Frequency ( $f$ ) — the number of complete cycles occurring per unit time, measured in hertz (Hz), where 1 Hz = 1 cycle per second.

T = \frac{1}{f} \qquad f = \frac{1}{T}

f = \frac{1}{T} \quad \text{or} \quad T = \frac{1}{f}

Worked Example

A pendulum completes 5 full swings in 10 seconds.

Frequency: $f = \dfrac{5 \text{ cycles}}{10\text{ s}} = 0.5\text{ Hz}$
Period: $T = \dfrac{1}{f} = \dfrac{1}{0.5} = 2\text{ s}$

Check: in 2 seconds you get one cycle, so in 10 seconds you get 5 cycles — consistent.

Where You'll Meet This

Context	What repeats	Typical period	Typical frequency
A tuning fork / concert pitch (the note A4)	Vibration of the prongs	≈ 0.00227 s	440 Hz
The note middle C on a piano	Air-pressure vibration	≈ 0.0038 s	≈ 262 Hz
AC electricity in India	Voltage polarity	0.02 s	50 Hz
Earth's rotation	Day and night	24 h	1/24 cycle per hour

Watch out

Don't confuse ordinary frequency $f$ with angular frequency $\omega$ . They're related by $\omega = 2\pi f$ , and $\omega$ is measured in radians per second, not hertz. You'll meet $\omega$ when studying oscillations and circular motion.

Why This Matters

Frequency and period are the basic language of anything that repeats — oscillations, waves, rotations, alternating circuits. When you read "50 Hz" for household AC supply, you now know the voltage completes 50 full cycles every second, each cycle lasting 0.02 s. When a problem gives "period = 2 s" for a pendulum, you know one full swing takes 2 seconds.

Key takeaway: period is time-per-cycle, frequency is cycles-per-time — reciprocals of each other, always.

If you've searched "Frequency And Period class 11 physics notes" or "Frequency And Period NCERT solutions", this page covers exactly that ground — the concept is a standard part of the Class 11 Physics NCERT/CBSE syllabus. It also carries real weight in JEE Main, NEET and state engineering/medical entrance exams, where questions on frequency and period test both conceptual understanding and calculation speed.

A function $f(t)$ is called periodic, with period $T$ , if it satisfies $f(t+T)=f(t)$ for every value of $t$ -- that is, it simply repeats its own pattern of values after every interval $T$ , with no restriction whatsoever on WHAT that repeating pattern actually looks like.

Not every periodic function describes S.H.M. S.H.M. is a much more specific, additional requirement: the restoring force (or, equivalently, the acceleration) at every instant must be directly proportional to the displacement from the mean position, and always directed opposite to it, $F=-kx$ or $a=-\omega^2x$ .

A clear counter-example is a small ball bouncing back and forth, at constant speed and with perfectly elastic collisions, between two rigid, fixed walls. This motion is certainly periodic: the ball's position repeats exactly after each round trip. But it is NOT S.H.M.: the ball's speed is constant between collisions (zero acceleration for most of the motion), and its velocity reverses instantaneously and discontinuously at each wall -- there is no smoothly varying restoring force proportional to displacement anywhere in this motion, so its displacement-time graph is a triangular (sawtooth-like) wave rather than a sine curve.

✓Final answer

f(t+T)=f(t)

Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

CBSE 2026Set ANNUAL1 markMCQ
$Q. What is the SI unit of frequency?$
(a) hertz (Hz)
(b) joule (J)
(c) watt (W)
(d) pascal (Pa)
›Reveal solutionSolution

Frequency = number of complete oscillations per unit time, and its SI unit is the hertz (Hz), equal to one cycle per second.

Frequency (f) of an oscillation or wave is defined as the number of complete cycles occurring per unit time:

f = 1/T (where T is the time period of one complete oscillation)

Its SI unit is the hertz (Hz), where 1 Hz = 1 cycle per second = 1 s^-1. (Joule is the unit of energy/work, watt is the unit of power, and pascal is the unit of pressure -- none of these measure a rate of repetition.)

✓Final answer
$(a) hertz (Hz).$
CBSE 2026Set ANNUAL1 mark
$Q. Write the relationship between angular frequency (ω) and time period (T).$
›Reveal solutionSolution

ω = 2π/T (angular frequency is 2π divided by the time period).

In one complete oscillation (one time period T), the phase of the motion advances by exactly 2π radians (a full cycle). Since angular frequency ω is defined as the rate of change of phase with time, ω = (total phase change)/(time taken) = 2π/T. Equivalently, since frequency f = 1/T (number of oscillations per second), this can also be written ω = 2πf.

✓Final answer
$ω = 2π/T.$
CBSE 2025Set ANNUAL1 markMCQ
$Q. The equation of displacement of a particle executing simple harmonic motion is y = sin(12πt + π/4). Frequency of the particle is (A) 8 Hz (B) 12 Hz (C) 6 Hz (D) 4 Hz$
›Reveal solutionSolution

The frequency of the given SHM is 6 Hz.

The given equation is $y = \sin(12\pi t + \pi/4)$ . Comparing with the standard SHM form $y = A\sin(\omega t + \phi)$ , the angular frequency is:

$\omega = 12\pi\text{ rad/s}$

Frequency is related to angular frequency by $\omega = 2\pi f$ :

$f = \dfrac{\omega}{2\pi} = \dfrac{12\pi}{2\pi} = 6\text{ Hz}$

✓Final answer
$(C) 6 Hz.$
CBSE 2022Set ANNUAL1 markMCQ
$Q. Match Column A item 'Time period' with the correct item in Column B.$
(a) kg m^2
(b) Poise
(c) Cp - Cv = R
(d) E = mc^2
(e) 1/frequency
(f) distance
(g) 24 hours
›Reveal solutionSolution

Time period T and frequency f are reciprocals of each other: T = 1/f.

Frequency f is defined as the number of complete oscillations (or cycles) per unit time, while the time period T is the time taken to complete one full oscillation. Since frequency counts oscillations per second and period measures seconds per oscillation, they are exact reciprocals: T = 1/f (and f = 1/T).

✓Final answer
$Time period — (e) 1/frequency.$

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Q.Define a periodic function mathematically. Are all periodic functions examples of simple harmonic motion? Justify your answer with an example of a motion that is periodic but is NOT S.H.M.

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