A particle is in linear simple harmonic motion between two points, A and B, 10 cm apart. Take the direction fr

Exercises · 13.5

Q. A particle is in linear simple harmonic motion between two points, A and B, 10 cm apart. Take the direction from A to B as the positive direction and give the signs of velocity, acceleration and force on the particle when it is

(a) at the end A,

(b) at the end B,

(d) at 2 cm away from B going towards A,

(e) at 3 cm away from A going towards B, and

(f) at 4 cm away from B going towards A.

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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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The key idea is that in Simple Harmonic Motion, the restoring force (and hence acceleration) always points toward the mean position (midpoint O), while velocity depends on the direction of motion.

Let O be the midpoint of AB, with A at $-5$ cm and B at $+5$ cm.

Step 1 – Position and force/acceleration:

Force and acceleration are opposite to displacement from O.

At A ( $-5$ cm): displacement is negative → force/acceleration is positive (toward B).
At B ( $+5$ cm): displacement is positive → force/acceleration is negative (toward A).
At O ( $0$ cm): displacement zero → force/acceleration = 0.

Step 2 – Velocity sign:

Velocity is positive when moving from A to B, negative when moving from B to A.

Step 3 – Apply to each case:

Case	Position	Direction	Velocity	Acceleration	Force
(a) at A	$-5$ cm	(turning)	0	+ (toward B)	+
(b) at B	$+5$ cm	(turning)	0	– (toward A)	–

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In SHM, the restoring force (and acceleration) always points toward the mean position, while velocity’s sign depends on the direction of motion. For a particle oscillating between A and B (A = –5 cm, B = +5 cm, mean at 0), the signs are: (a) A: v=0, a>0, F>0;

(b) B: v=0, a<0, F<0;

(d) 2 cm from B going to A: v<0, a<0, F<0; (e) 3 cm from A going to B: v>0, a<0, F<0; (f) 4 cm from B going to A: v<0, a<0, F<0.

The core idea

Simple harmonic motion is defined by a restoring force proportional to displacement from the mean position, always directed toward the mean. That means acceleration and force share the same sign — opposite to the displacement. Velocity, however, depends purely on which way the particle is moving at that instant.

We place the mean position (equilibrium) at the centre of AB. Since A and B are 10 cm apart, the amplitude is 5 cm. Let’s set A at $x = -5$ cm and B at $x = +5$ cm, with the positive direction from A to B.

x = 0

Step-by-step reasoning

1. (a) At end A ( $x = -5$ cm)

The particle is at the extreme left. It is momentarily at rest before turning back toward the mean.

Velocity: zero (turning point).
Displacement is negative ( $x = -5$ ). Force and acceleration are opposite to displacement: $F = -k(-5) = +5k$ , so positive.
Signs: $v = 0$ , $a > 0$ , $F > 0$ .

2. (b) At end B ( $x = +5$ cm)

The particle is at the extreme right, momentarily at rest.

Velocity: zero.
Displacement positive, so force and acceleration are negative.
Signs: $v = 0$ , $a < 0$ , $F < 0$ .

3. (c) At mid-point ( $x = 0$ ) going toward A

The mean position has zero displacement, so force and acceleration are zero. The particle is moving left (negative direction).

Velocity: negative.
Signs: $v < 0$ , $a = 0$ , $F = 0$ .

4. (d) At 2 cm away from B going toward A

B is at $+5$ cm, so 2 cm away from B means $x = +3$ cm (since going toward A means moving left from B). The particle is on the positive side, moving left.

Displacement positive → force and acceleration negative.
Velocity negative (moving toward A).
Signs: $v < 0$ , $a < 0$ , $F < 0$ .

5. (e) At 3 cm away from A going toward B

A is at $-5$ cm, so 3 cm away from A means $x = -2$ cm (since going toward B means moving right from A). The particle is on the negative side, moving right.

Displacement negative → force and acceleration positive. …

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Q.A particle is in linear simple harmonic motion between two points, A and B, 10 cm apart. Take the direction from A to B as the positive direction and give the signs of velocity, acceleration and force on the particle when it is

The core idea

Step-by-step reasoning

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