The rotation of earth about its axis is (Note: more than one of the given options may be correct.) (a) periodi

NCERT Exemplar · Q12

Q. The rotation of earth about its axis is (Note: more than one of the given options may be correct.)

(a) periodic motion.

(b) simple harmonic motion.

(d) non-periodic motion.

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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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Earth's rotation repeats at regular intervals, making it a periodic motion. However, it is a continuous rotation, not an oscillation about an equilibrium position, and lacks a restoring force proportional to displacement, so it is not simple harmonic motion. Therefore, Earth's rotation is periodic but not simple harmonic.

To understand the nature of Earth's rotation, we first need to define and distinguish between periodic motion and simple harmonic motion (SHM).

1. Understanding Periodic Motion

Periodic motion is any motion that repeats itself after a fixed interval of time. The time taken for one complete repetition is called the period.

Key characteristic: Repetition at regular time intervals.
Examples:
- The revolution of Earth around the Sun (period $\approx 365.25$ days).
- The hands of a clock moving.
- A child on a swing.
- The motion of a planet in its orbit.

2. Understanding Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a special type of oscillatory (to and fro) periodic motion. For a motion to be classified as SHM, two primary conditions must be met:

Oscillatory Nature: The object must move back and forth about a fixed equilibrium position.
Restoring Force/Torque: There must be a restoring force (or torque for rotational motion) that is directly proportional to the displacement from the equilibrium position and always directed towards that equilibrium position.
- Mathematically, for linear SHM, the restoring force $F$ is given by $F = -kx$ , where $k$ is a positive constant and $x$ is the displacement from equilibrium. The negative sign indicates that the force is always opposite to the displacement.
- For angular SHM, the restoring torque $\tau$ is given by $\tau = -k\theta$ , where $\theta$ is the angular displacement.

Important

All simple harmonic motions are periodic, but not all periodic motions are simple harmonic. SHM is a specific type of periodic motion with additional constraints on the nature of the restoring force and the oscillatory path.

3. Analyzing Earth's Rotation

Now, let's apply these concepts to the rotation of Earth about its axis.

Is Earth's rotation periodic motion?

Yes. Earth completes one full rotation about its axis approximately every 24 hours (one sidereal day). This motion repeats consistently after a fixed interval of time. Therefore, Earth's rotation is a periodic motion.
Is Earth's rotation simple harmonic motion?

No. Let's check the conditions for SHM:
- Oscillatory Nature: Earth's rotation is a continuous rotation in one direction, not a back-and-forth oscillation about an equilibrium angular position. It does not swing to one side, stop, reverse direction, swing to the other side, and so on. …

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Q.The rotation of earth about its axis is (Note: more than one of the given options may be correct.)

1. Understanding Periodic Motion

2. Understanding Simple Harmonic Motion (SHM)

3. Analyzing Earth's Rotation

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