The equation of motion of a particle is x = acos(α t)². The motion is (a) periodic but not oscillatory. (b) pe

NCERT Exemplar · Q9

x = a\cos(\alpha t)^{2}

(a) periodic but not oscillatory.

(b) periodic and oscillatory.

(d) neither periodic nor oscillatory.

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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). …

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Using the identity $\cos^2\theta = \dfrac{1+\cos2\theta}{2}$ , the motion $x=a\cos^2(\alpha t)$ rewrites as $x=\dfrac a2+\dfrac a2\cos(2\alpha t)$ -- a cosine oscillation about the mean position $x=\dfrac a2$ , repeating every $T=\dfrac{\pi}{\alpha}$ . Since the particle both repeats regularly and reverses direction about a fixed mean, the motion is (B) periodic and oscillatory.

Two different questions to ask

Periodic motion simply repeats itself after a fixed time interval: $x(t+T)=x(t)$ . Oscillatory motion additionally requires the particle to move back and forth -- reversing direction -- about some mean (equilibrium) position. A particle going around a circle at constant speed is periodic but not oscillatory (it never reverses); a pendulum is both.

Rewriting the equation

Apply the double-angle identity $\cos^2\theta=\dfrac{1+\cos2\theta}{2}$ with $\theta=\alpha t$ :

x = a\cos^2(\alpha t) = a\cdot\frac{1+\cos(2\alpha t)}{2} = \frac a2 + \frac a2\cos(2\alpha t)

This is now transparent: $x$ is a cosine function with amplitude $\dfrac a2$ , angular frequency $2\alpha$ , oscillating about the mean value $\dfrac a2$ .

Checking periodicity

The term $\cos(2\alpha t)$ repeats with period

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

so $x(t+T)=x(t)$ for all $t$ -- the motion is periodic.

Checking oscillatory behaviour …

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Q.The equation of motion of a particle is x=acos⁡(αt)2x = a\cos(\alpha t)^{2}x=acos(αt)2. The motion is

Two different questions to ask

Rewriting the equation

Checking periodicity

Checking oscillatory behaviour …

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Q.The equation of motion of a particle is $x = a\cos(\alpha t)^{2}$ . The motion is

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