Q.Which of the following examples represent periodic motion?

Concept: Periodic motion repeats itself at regular intervals of time.

A motion is periodic if the body returns to its initial position and velocity after a fixed time interval called the period. The key test: does the motion repeat identically, over and over?

Analysis of each case:

1. The swimmer completes one return trip and stops — no repetition occurs. This is non-periodic motion.
2. A freely suspended bar magnet, when displaced and released, oscillates about the N-S direction due to Earth's magnetic field. It repeatedly swings back and forth through the equilibrium position with a constant period. This is periodic (simple harmonic motion with damping in practice, but fundamentally periodic).
3. A hydrogen molecule rotating about its center of mass completes identical rotations continuously. Each rotation takes the same time, and the motion repeats indefinitely. This is periodic.
4. An arrow released from a bow follows a parabolic trajectory and does not return to repeat its motion. This is non-periodic.
   ✓Final answer

   The examples representing periodic motion are (b) and (c).

Understanding Periodic Motion

Periodic motion is motion that repeats itself in equal intervals of time, tracing the same path over and over. The key requirements are:

• The motion must return to its starting configuration
• It must do so repeatedly, not just once
• The time interval between repetitions (the period) must be constant

Think of a pendulum, a planet orbiting the Sun, or your heartbeat — each completes a cycle and immediately begins the next identical cycle.

Now let's examine each scenario:

1. The swimmer (a) — NOT periodic

   The swimmer completes one return trip and stops. Yes, the motion involves returning to the starting point, but that's where it ends. There's no repetition, no second cycle, no third. The swimmer doesn't automatically begin another lap the moment they touch the bank.

   Periodic motion requires the cycle to continue indefinitely (or at least for multiple cycles). A single back-and-forth journey is just that — a single journey, not a repeating pattern.

2. The bar magnet (b) — PERIODIC

   When you displace a freely suspended bar magnet from its North-South equilibrium and release it, Earth's magnetic field exerts a restoring torque. The magnet swings back toward the N-S line, overshoots due to inertia, swings to the other side, and returns again.

   This is simple harmonic motion (or damped harmonic motion if we account for air resistance). The magnet oscillates about the N-S direction repeatedly with a definite period. Each swing follows the same path, and the motion continues cycle after cycle until friction eventually damps it out — but the nature of the motion is periodic.

   Tip

   Any system with a restoring force proportional to displacement (or restoring torque proportional to angular displacement) executes periodic motion, at least approximately.

3. The hydrogen molecule (c) — PERIODIC

   A hydrogen molecule ($\text{H}_2$) rotating about its center of mass is a beautiful example of periodic motion. The molecule spins continuously, and after each complete rotation of angle $2\pi$, it returns to exactly the same configuration.

   The period is $T = \frac{2\pi}{\omega}$, where $\omega$ is the angular velocity. This rotation continues indefinitely (ignoring quantum effects and collisions), making it perfectly periodic. Every point on the molecule traces a circular path repeatedly.

4. The arrow (d) — NOT periodic

   Once released from a bow, the arrow follows a parabolic trajectory under gravity and eventually hits the ground (or a target). It does not return to the bow, it does not repeat its path, and it certainly doesn't launch itself again.

   This is projectile motion — a one-time event, not a repeating cycle.

Step 1: A motion is periodic only if it repeats itself, identically, indefinitely at a fixed time interval — a single event, however symmetric, is not periodic.

Step 2 (a): The swimmer's one return trip ends when they touch the bank; nothing repeats after that, so it is non-periodic.

Step 3 (b): A bar magnet displaced from the N–S line and released oscillates about that line under Earth's restoring magnetic torque, repeating the same swing indefinitely — periodic.

Step 4 (c): A rotating $\text{H}_2$ molecule returns to an identical orientation every $2\pi$ of rotation and keeps rotating — periodic.

Step 5 (d): The arrow follows a single parabolic flight and never repeats it on its own — non-periodic.