Show that for a particle executing S.H.M, velocity and displacement have a phase difference of π/2.

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\pi/2

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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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In simple harmonic motion, the velocity is the time derivative of displacement. Since displacement varies as $\sin(\omega t + \phi)$ and velocity as $\cos(\omega t + \phi)$ , the two functions are shifted by a quarter of a cycle — a phase difference of $\pi/2$ radians.

The core idea: why phase matters

Simple harmonic motion is the projection of uniform circular motion onto a diameter. If you watch a point moving on a circle at constant angular speed $\omega$ , its horizontal position oscillates back and forth. That oscillation is SHM.

The key insight: the velocity of the particle is not in step with its displacement. When the particle is at the extreme end (maximum displacement), it is momentarily at rest — velocity is zero. When it passes through the centre (zero displacement), it is moving fastest. That lag between position and speed is exactly a quarter-cycle, or $\pi/2$ in phase.

Let's make this precise.

Step-by-step derivation

Write the displacement equation. For a particle executing SHM, the displacement from the mean position is given by

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

where $A$ is the amplitude, $\omega$ the angular frequency, $t$ the time, and $\phi$ the initial phase (or phase constant).

This is the standard form — you could also use a cosine, but the phase difference we find will be independent of that choice.

Find the velocity by differentiation. Velocity is the rate of change of displacement:

v = \frac{dx}{dt} = A \omega \cos(\omega t + \phi)

Rewrite the velocity in sine form. Using the identity $\cos\theta = \sin(\theta + \pi/2)$ , we get

v = A \omega \sin\!\left(\omega t + \phi + \frac{\pi}{2}\right)

Compare the two expressions.
- Displacement: $x = A \sin(\omega t + \phi)$
- Velocity: $v = A \omega \sin\!\left(\omega t + \phi + \frac{\pi}{2}\right)$
The velocity has the same sinusoidal form as displacement, but its argument (the phase) is larger by $\pi/2$ . That extra $\pi/2$ is the phase difference. …

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Q.Show that for a particle executing S.H.M, velocity and displacement have a phase difference of π/2\pi/2π/2.

The core idea: why phase matters

Step-by-step derivation

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Q.Show that for a particle executing S.H.M, velocity and displacement have a phase difference of $\pi/2$ .

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