Q.A hollow sphere is filled with water. It is hung by a long thread. As the water flows out of a hole at the bottom, the period of oscillation will a) first increase and then decrease b) first decrease and then increase c) increase continuously d) decrease continuously

Simple Pendulum Period: From Intuition to Formula

Imagine tying a small weight to a string, holding the other end fixed, and giving it a gentle push. It swings back and forth — that’s a simple pendulum. The question is: what determines how fast it swings? Does a heavier bob swing faster? Does a longer string make it slower?

Let’s start with what you already feel. If you hold a short string (say 20 cm) and swing it, the bob zips back and forth quickly. If you use a long string (say 1 m), the swing is noticeably slower. So length matters — longer means slower.

What about the weight? Try a light plastic bob and a heavy metal one of the same size, on the same string. You’ll find they swing at the same speed. That’s surprising — heavier things don’t fall faster, and here they don’t swing faster either. So mass does not affect the period (the time for one complete back-and-forth swing).

What about how hard you push? If you give a big push, the bob swings wider, but does it take more time? For small swings (small angles, say less than about 15°), the period is almost the same regardless of amplitude. That’s the key: for small oscillations, the pendulum is isochronous — its period is independent of amplitude.

The Precise Statement

For a simple pendulum of length $L$ (measured from pivot to centre of bob), swinging with small amplitude in a uniform gravitational field $g$, the time period $T$ (time for one complete oscillation) is:

That’s it. No mass term. No amplitude term (for small angles).

Why does this formula make sense?

• $L$ in numerator: longer string → larger $T$ (slower swing). Doubling $L$ multiplies $T$ by $\sqrt{2} \approx 1.4$.
• $g$ in denominator: stronger gravity (larger $g$) → smaller $T$ (faster swing). On the Moon ($g \approx 1.6$ m/s²), the same pendulum swings much slower.
• $2\pi$: comes from the mathematics of simple harmonic motion — the pendulum’s motion is approximately sinusoidal for small angles.

Common exam pitfalls

• Don’t confuse $L$ with amplitude. $L$ is the string length, not how far you pull it. …