Q.The period of oscillations of a simple pendulum increases by 10%, when its length is increased by 21 cm. Find its initial length and initial period.

A simple pendulum is a heavy bob suspended by a light, inextensible string from a rigid support. For small angular displacements theta (below about 10 degrees), the restoring force on the bob, F = -mgsin(theta), can be approximated (using sin(theta) is approximately theta, and theta is approximately x/L, where x is the arc-length displacement and L the string length) as F = -(mg/L)x -- directly proportional to displacement and oppositely directed, exactly the S.H.M. condition. This gives the acceleration-per-unit-displacement as g/L, and hence the period T = 2pisqrt(L/g), derived under three assumptions: small amplitude, a long string, and motion confined to a single vertical plane. The corresponding "laws of the simple pendulum": period is proportional to sqrt(L), inversely proportional to sqrt(g), independent of the bob's mass, and independent of amplitude (for small amplitude).

A second's pendulum is one whose period is exactly 2 seconds, giving the length-gravity relation L_s = g/pi^2 -- usable to find g from a measured length, or to predict the length needed for a given g. Experimentally, the small-angle approximation breaks down gradually as amplitude grows: the period is roughly 2% too high at 20 degrees amplitude, 5% at 50 degrees, 10% at 70 degrees, and 18% at 90 degrees, which is why the recommended maximum amplitude for a simple-pendulum experiment is kept under 20 degrees. A simple pendulum differs from a conical pendulum (studied under circular motion) in almost every respect -- plane of motion (vertical arc vs. horizontal circle), energy behaviour (kinetic and potential interconvert vs. both stay separately constant), the governing force component, and the period formula itself, T=2pisqrt(L/g) versus the conical pendulum's T=2pisqrt(L*cos(theta)/g).