Simple Pendulum (Period)

Insert the pendulum length in metres into T = 2π·√(l/g), with g = 9.81 m/s². Example: a 1 m pendulum has T = 2π·√(1/9.81) = 2π × 0.319 ≈ 2.0 s. The length is measured from the pivot to the centre of mass of the bob, not to the end of the string. Note that T grows only with √l under the root: for twice the period you need four times the length. Mass and amplitude deliberately do not appear in the formula; for small angles they have no measurable influence. Exactly this makes the pendulum a reliable timekeeper.

The mass plays a double role in the pendulum that cancels exactly. On the one hand the restoring force is proportional to the mass, being a component of the weight m·g·sinφ. On the other hand the inertia that must be accelerated is also proportional to the mass. In the equation of motion m·a = −m·g·sinφ the m cancels, leaving an equation containing only l and g. This is the same reason why all bodies fall equally fast. The spring pendulum is different: there the restoring spring force is independent of the mass, which is why T = 2π·√(m/D) does contain the mass.

Rearrange the formula for g: g = 4π²·l/T². Measure the pendulum length as precisely as possible to the centre of the bob and time 10 or 20 full oscillations to reduce the stopwatch error; then divide by the count. Example: l = 0.8 m and T = 1.8 s give g = 39.48 × 0.8/3.24 ≈ 9.75 m/s². Keep the amplitude below about 10°, otherwise T becomes systematically too large and g too small. The method is astonishingly precise: with careful length measurement, school experiments can come within one percent of the literature value 9.81 m/s².

The period exactly doubles, because T grows with the square root of the length: √4 = 2. A pendulum with T = 1 s (l ≈ 0.25 m) becomes one with T = 2 s (l ≈ 0.994 m), the classic seconds pendulum, which takes exactly one second per half swing. Conversely you must quarter the length to double the frequency. This square-root dependence also explains why long swings move leisurely and short ones frantically. To correct the period by only a few percent, for instance when regulating a pendulum clock, you therefore shift the bob only minimally up or down.

The formula rests on the small-angle approximation sin φ ≈ φ and therefore holds only for displacements up to about 10°. At larger amplitudes the pendulum swings measurably slower: at 30° the period is already about 1.7 % longer, at 90° about 18 %. The model also assumes a massless, inextensible string and a point mass; an extended rigid body is a physical pendulum with T = 2π·√(J/(m·g·d)) and moment of inertia J. Air resistance and pivot friction damp the oscillation but change the period only slightly. Precision measurements must correct for all three effects.