Only two forces act on the bob of a pendulum: gravity straight down, and the tension in the string.
Gravity points straight down, but the bob can only move along a circle. The direction of the force and the direction it can move cross at an angle, so gravity is split in two: along the string, and at right angles to it. The right-angled one is the direction the bob travels.
The component along the string, , is at right angles to the direction of travel, and a force at right angles does not change the speed. The tension lies along the same direction, so every force in that direction is at work holding the bob on its circle.
What is left, , moves the bob. It always points toward the lowest point, opposite to the side the bob has swung to, so the force along the tangent is . As with the restoring force of a spring, the minus sign means points the other way.
With a spring, the displacement was measured along a straight line. The bob of a pendulum moves along a circle, so its displacement is measured along the circle too. Take to be the length of arc from the lowest point to the bob.
It is the arc of a circle of radius subtending an angle , so , with measured in radians. Measuring an angle in radians was, from the start, measuring it by arc length.
The materials are ready. The force is and the displacement is , both written with . But the two are not yet proportional, because and are different things. To be called simple harmonic motion, the force has to be proportional to the displacement.
Draw and on the same axes. is a straight line; is a curve that flattens off further along.
Near the origin, however, the two nearly coincide. While the swing is small, either one gives the same answer, so we set . That is the only approximation anywhere in the argument; everything else is exact.
The force then becomes . Since , we have , and putting that in gives . Force proportional to displacement, opposite in direction: exactly the form of the spring's , with playing the part of . Motion is not simple harmonic because it is a spring; it is simple harmonic because it takes this form. A pendulum takes this form only when it swings through small angles.
The period of simple harmonic motion was . With in the part of , the two cancel under the root and .
The mass has gone. Heavy bob or light, the same length of string gives the same period. The reason is plain: the force pulling the bob, , and the reluctance of the bob to be moved, , are both proportional to the same . Make it heavier and it is pulled harder, but it is just as much harder to move, and the two cancel.
The amplitude does not appear either. Large swing or small, one round trip takes the same time. The two pendulums in the figure differ in weight and in swing, yet they reach their ends together and return together. This is called isochronism. It holds under , though, and a swing too large for the approximation has a period longer than this equation gives.