Which of the following relationships between the acceleration a and the displacement x of a particle involve s

Exercises · 13.6

a

(a)

a = 0.7x

(b)

a = -200x^{2}

(c)

a = -10x

(d)

a = 100x^{3}

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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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Simple harmonic motion requires acceleration to be directly proportional to negative displacement ( $a \propto -x$ ). Only option (c) $a = -10x$ satisfies this condition, so it is the correct answer.

The Core Idea: What Makes Motion "Simple Harmonic"?

Simple harmonic motion (SHM) is not just any back-and-forth motion. It has a very specific mathematical fingerprint: the acceleration of the particle must be directly proportional to its displacement from a fixed point, and the acceleration must always point toward that fixed point (i.e., opposite to the displacement).

Why this condition? Think of a mass on a spring. When you pull it to the right (positive $x$ ), the spring pulls it back to the left (negative $a$ ). The harder you pull, the stronger the restoring force — that's the proportionality. This relationship produces a sinusoidal motion in time, which is the hallmark of SHM.

a = -\omega^2 x

Now, let's test each option against this standard.

Step-by-Step Analysis

1. Option (a): $a = 0.7x$

Here, the acceleration is positive when $x$ is positive. That means if the particle moves to the right, the acceleration also pushes it to the right — it speeds away from the centre, not back toward it. This is a repulsive force, not a restoring one. The negative sign is missing, so this cannot be SHM.

2. Option (b): $a = -200x^2$

This has the correct negative sign, so the acceleration points toward the centre. But look at the exponent: $x$ is squared. The acceleration is proportional to $x^2$ , not $x$ . For SHM, the relationship must be linear — doubling $x$ must double $a$ (in magnitude). Here, doubling $x$ quadruples $a$ . This non-linear relationship produces a motion that is not sinusoidal; it's a different kind of oscillation. So this is not SHM.

Watch out

A common mistake is to think any equation with a negative sign and a power of $x$ is SHM. The power must be exactly 1 — no squares, no cubes, no square roots.

3. Option (c): $a = -10x$ …

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Q.Which of the following relationships between the acceleration aaa and the displacement xxx of a particle involve simple harmonic motion?

The Core Idea: What Makes Motion "Simple Harmonic"?

Step-by-Step Analysis

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Q.Which of the following relationships between the acceleration $a$ and the displacement $x$ of a particle involve simple harmonic motion?

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