The displacement of a particle is represented by the equation $y = 3\cos\left(\dfrac{\pi}{4} - 2\omega t\right)$. The motion of the particle is (a) simple harmonic with period 2p/w. (b) simple harmonic with period $\pi/\omega$. (c) periodic but not simple harmonic. (d) non-periodic.

> [!TLDR] > A displacement of the form $y = A\cos(\alpha - \omega't)$ is simple harmonic motion; rewriting reveals angular frequency $2\omega$ and hence period $\frac{2\pi}{2\omega} = \frac{\pi}{\omega}$. ## Why this is simple harmonic motion Simple harmonic motion (SHM) requires the displacement to be a sinusoidal function of time. The standard forms are $y = A\sin(\omega't + \phi)$ or $y = A\cos(\omega't + \phi)$, where $A$ is amplitude, $\omega'$ is angular frequency, and $\phi$ is a phase constant. The given equation $y = 3\cos\left(\frac{\pi}{4} - 2\omega t\right)$ might look unusual because of the subtraction inside the cosine, but cosine is an even function: $\cos(-\theta) = \cos(\theta)$. This means we can rewrite the argument. ## Step-by-step analysis 1. **Rewrite the displacement equation** Start with $y = 3\cos\left(\frac{\pi}{4} - 2\omega t\right)$. Factor out the negative sign: $$y = 3\cos\left[-\left(2\omega t - \frac{\pi}{4}\right)\right]$$ Using $\cos(-\theta) = \cos(\theta)$: $$y = 3\cos\left(2\omega t - \frac{\pi}{4}\right)$$ 2. **Identify the form** This is now clearly in the standard SHM form $y = A\cos(\omega't + \phi)$, where: - Amplitude $A = 3$ - Angular frequency $\omega' = 2\omega$ - Phase constant $\phi = -\frac{\pi}{4}$ 3. **Calculate the period** The period $T$ of SHM is related to angular frequency by: $$T = \frac{2\pi}{\omega'}$$ Substituting $\omega' = 2\omega$: $$T = \frac{2\pi}{2\omega} = \frac{\pi}{\omega}$$ > [!WARNING] > Don't confuse the parameter $\omega$ in the problem with the angular frequency of the motion. The angular frequency is $2\omega$, not $\omega$. > [!FORMULA] > $$T = \frac{2\pi}{\text{angular frequency}} = \frac{2\pi}{2\omega} = \frac{\pi}{\omega}$$ The motion is indeed simple harmonic (a pure cosine function of time) with period $\pi/\omega$. > [!ANSWER] > The correct option is **(B)**: simple harmonic with period $\pi/\omega$.

The displacement of a particle is represented by t – Class 11 Physics

NCERT Exemplar · Q1

y = 3\cos\left(\dfrac{\pi}{4} - 2\omega t\right)

(a) simple harmonic with period 2p/w.

(b) simple harmonic with period

\pi/\omega

(d) non-periodic.

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Concept understanding — Simple Harmonic Motion

Simple Harmonic Motion: The Natural Rhythm of Things

Imagine a ball placed at the bottom of a perfectly smooth, U-shaped bowl. If you give it a gentle push, what happens? It rolls up one side, slows down, stops for an instant, then rolls back down, past the bottom, up the other side, stops, and returns. Left alone, it keeps doing this — back and forth, back and forth — in a steady, repeating rhythm.

That rhythm is the heart of Simple Harmonic Motion (SHM). It's the most fundamental kind of oscillatory (back-and-forth) motion in physics.

The Intuition: A Restoring Force That Fights Displacement

The key idea is this: the further you push the object from its resting (equilibrium) position, the stronger the force that tries to pull it back.

In the bowl, when the ball is at the bottom (equilibrium), gravity pulls straight down, and the bowl pushes straight up — no sideways force. But when you push the ball up the side, gravity now has a component that pulls it down the slope. The higher up the side you push it, the steeper the slope, and the stronger that pull-back force becomes.

This is a restoring force — it always points toward equilibrium. And crucially, in SHM, this restoring force is directly proportional to the displacement from equilibrium. Double the displacement, double the restoring force.

F = -k x

The Precise Statement

Simple Harmonic Motion is the motion of an object where the restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction.

That's it. That single condition — $F = -kx$ — is the entire definition. Everything else (the sine waves, the formulas for period and frequency) follows mathematically from this one law.

What Does This Motion Look Like?

If you track the ball's position over time, you get a beautiful, smooth wave — a sine wave (or cosine wave). It's the same shape as the shadow of a spinning wheel cast on a wall.

The motion has three key descriptors:

Amplitude ( $A$ ): The maximum displacement from equilibrium. How far you initially pushed the ball up the side of the bowl.
Period ( $T$ ): The time it takes to complete one full back-and-forth cycle (e.g., from the leftmost point, back to the leftmost point).
Frequency ( $f$ ): How many cycles happen per second. $f = 1/T$ .

Note

A remarkable fact: for a given system (fixed $k$ and fixed mass $m$ ), the period and frequency do not depend on the amplitude. A big push and a tiny push take exactly the same time to complete one cycle. This is called isochronism — and it's why pendulums were used to keep time in clocks.

The Mathematical Description (Derived from $F = -kx$ )

Using Newton's second law ( $F = ma$ ) and the definition of acceleration ( $a = \frac{d^2x}{dt^2}$ ), the condition $F = -kx$ becomes:

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

This is a differential equation. Its solution — the position as a function of time — is:

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

Where:

$\omega = \sqrt{\frac{k}{m}}$ is the angular frequency (radians per second). It tells you how fast the oscillation is.
$\phi$ is the phase constant (determines where in the cycle you start measuring time).

From $\omega$ , you get the period: $T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{m}{k}}$ .

Watch out

Do not confuse angular frequency $\omega$ (rad/s) with ordinary frequency $f$ (Hz). They are related by $\omega = 2\pi f$ . Many exam errors come from mixing these up.

Real-World Examples

SHM is an idealization — a perfect model. But many real systems approximate it beautifully:

A mass on a spring (horizontal or vertical) — the classic textbook example.
A simple pendulum — but only for small angles (less than about $15^\circ$ ). For large swings, the restoring force is no longer proportional to displacement, and the motion is not simple harmonic.
The vibration of atoms in a solid — each atom is held in place by bonds that act like tiny springs.
A tuning fork — the prongs vibrate in SHM, producing a pure tone.

The Bottom Line

Simple Harmonic Motion is any motion driven by a restoring force that is proportional to and opposite the displacement. It produces a sinusoidal oscillation with a constant period that is independent of amplitude. Everything else — the equations, the graphs, the energy transformations — is just unpacking that single, elegant idea.

Looking up "Simple Harmonic Motion: Definition, Formula & Real-World Examples" or "Simple Harmonic Motion important questions 11" is a common way students land here, and rightly so — simple harmonic motion is a core part of the Class 11 Physics NCERT/CBSE curriculum. Expect it to reappear, often in a slightly disguised form, across JEE Main, NEET and state engineering/medical entrance exams.

A displacement of the form $y = A\cos(\alpha - \omega't)$ is simple harmonic motion; rewriting reveals angular frequency $2\omega$ and hence period $\frac{2\pi}{2\omega} = \frac{\pi}{\omega}$ .

Why this is simple harmonic motion

Simple harmonic motion (SHM) requires the displacement to be a sinusoidal function of time. The standard forms are $y = A\sin(\omega't + \phi)$ or $y = A\cos(\omega't + \phi)$ , where $A$ is amplitude, $\omega'$ is angular frequency, and $\phi$ is a phase constant.

The given equation $y = 3\cos\left(\frac{\pi}{4} - 2\omega t\right)$ might look unusual because of the subtraction inside the cosine, but cosine is an even function: $\cos(-\theta) = \cos(\theta)$ . This means we can rewrite the argument.

Step-by-step analysis

Rewrite the displacement equation Start with $y = 3\cos\left(\frac{\pi}{4} - 2\omega t\right)$ . Factor out the negative sign:

y = 3\cos\left[-\left(2\omega t - \frac{\pi}{4}\right)\right]

Using $\cos(-\theta) = \cos(\theta)$ :

y = 3\cos\left(2\omega t - \frac{\pi}{4}\right)

Identify the form

This is now clearly in the standard SHM form $y = A\cos(\omega't + \phi)$ , where:
- Amplitude $A = 3$
- Angular frequency $\omega' = 2\omega$
- Phase constant $\phi = -\frac{\pi}{4}$
Calculate the period

The period $T$ of SHM is related to angular frequency by:

T = \frac{2\pi}{\omega'}

Substituting $\omega' = 2\omega$ :

T = \frac{2\pi}{2\omega} = \frac{\pi}{\omega}

Watch out

Don't confuse the parameter $\omega$ in the problem with the angular frequency of the motion. The angular frequency is $2\omega$ , not $\omega$ .

T = \frac{2\pi}{\text{angular frequency}} = \frac{2\pi}{2\omega} = \frac{\pi}{\omega}

The motion is indeed simple harmonic (a pure cosine function of time) with period $\pi/\omega$ .

✓Final answer

\pi/\omega

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

Council of Higher Secondary Education, Manipur (Higher Secondary 1st Year) 2026Set ANNUAL1 markMCQ
$Q. Which of the following function of time represent periodic motion?$
(a) sin ωt + cos ωt
(b) e^(-iωt)
(c) e^(iωt)
(d) log ωt
›Reveal solutionSolution
sin ωt + cos ωt represents periodic motion; the others do not represent physically valid periodic motion.

A function f(t) represents periodic motion if it is a real, bounded, physically meaningful function of time satisfying f(t + T) = f(t) for some fixed T.

sin ωt + cos ωt: this is a real, bounded function; it can be written as √2 sin(ωt + π/4), which clearly repeats with period T = 2π/ω. This is genuine periodic motion (in fact, SHM).

e^(−iωt) and (c) e^(iωt): these are complex-valued functions. While they are mathematically periodic (by Euler's formula, e^(±iω(t+2π/ω)) = e^(±iωt)), a physical displacement must be a real quantity — a complex exponential does not by itself represent the position of a real oscillating body, so it is not treated as representing physical periodic motion in this context.

(d) log ωt: this grows without bound as t increases and never repeats its values — it is not periodic at all.

✓Final answer
$The correct option is (a) sin ωt + cos ωt.$
Council of Higher Secondary Education, Manipur (Higher Secondary 1st Year) 2025Set ANNUAL1 markMCQ
$Q. Assertion (A): A uniform circle motion is a simple harmonic motion. Reason (R): The projection of uniform circular motion on any diameter of a circle of reference is simple harmonic motion.$
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
(b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
(c) Assertion (A) is true, but Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is also false.
›Reveal solutionSolution

The projection of uniform circular motion on any diameter is genuinely simple harmonic motion (Reason is a true, standard NCERT fact) and is exactly the basis for the well-known link between circular motion and SHM stated in the Assertion.

Reason (R): "The projection of uniform circular motion on any diameter of a circle of reference is simple harmonic motion" — this is a precisely correct, standard NCERT result. If a particle P moves with constant angular speed $\omega$ around a circle of radius $A$ , the foot of the perpendicular from P onto any diameter (its projection) oscillates back and forth with displacement $x(t) = A\cos(\omega t + \phi)$ , which is exactly the equation of SHM.

Assertion (A): as commonly used in Class 11 question banks, this statement is intended to express the classic textbook connection between uniform circular motion and SHM (that SHM can be understood/derived via uniform circular motion) — with R giving the precise mechanism (the projection) that establishes this connection.

A precision note for the student: strictly speaking, uniform circular motion itself (2-D motion, constant speed, constantly changing direction) is not identical to SHM (1-D oscillatory motion with acceleration proportional to displacement) — it is the projection of UCM onto a diameter that is SHM. Keep this distinction in mind for a written/derivation question; for this Assertion-Reason format, R correctly explains and grounds the well-known link the Assertion is pointing at.

✓Final answer
$The correct option is (a) — both A and R are true, and R (the projection of UCM on a diameter is SHM) is the correct explanation for the connection between circular motion and SHM being asserted.$
Council of Higher Secondary Education, Manipur (Higher Secondary 1st Year) 2023Set ANNUAL1 mark
$Q. What is the minimum condition for a system to execute S.H.M.?$
›Reveal solutionSolution
A system executes SHM only if the force pulling it back to equilibrium is proportional to displacement and always points toward the mean position: $F = -kx$ .

Simple harmonic motion (SHM) is a special type of periodic oscillatory motion. For a system to execute SHM, there must exist a restoring force (or, equivalently, a restoring torque for angular SHM) that satisfies two conditions simultaneously:

The magnitude of the restoring force must be directly proportional to the magnitude of the displacement $x$ of the body from its mean (equilibrium) position: $F \propto x$ .

The restoring force must always act in the direction opposite to the displacement, i.e., it always tries to bring the body back toward the mean position.

Together these give $F = -kx$ , where $k$ is a positive constant (called the force constant). Correspondingly, the acceleration is $a = -\omega^2 x$ , where $\omega^2 = k/m$ . If either condition fails — for example if the restoring force is not linear in $x$ , or does not always oppose the displacement — the resulting motion, even if periodic, is not simple harmonic.

✓Final answer
$F = -kx$

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Q.The displacement of a particle is represented by the equation y=3cos⁡(π4−2ωt)y = 3\cos\left(\dfrac{\pi}{4} - 2\omega t\right)y=3cos(4π​−2ωt). The motion of the particle is

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Step-by-step analysis

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Q.The displacement of a particle is represented by the equation $y = 3\cos\left(\dfrac{\pi}{4} - 2\omega t\right)$ . The motion of the particle is

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