Q.Two identical springs of constant k are connected, first in series and then in parallel. A metal block of mass m is suspended from their combination. The ratio of their frequencies of vertical oscillations will be in a ratio (A) 1:4
(B) 1:2
(C) 2:1
(D) 4:1

The amplitude A of an S.H.M. is its maximum displacement from the mean position -- the value x reaches when sin(omega t + phi) = +/-1. The period T is the time for one complete oscillation; since x, v and a in the general expression x = A sin(omega t + phi) all return to their original values whenever the phase increases by exactly 2pi, the period is T = 2pi/omega. For a spring-block oscillator, since omega^2 = k/m, this becomes the practical formula T = 2pisqrt(m/k). More generally, for ANY S.H.M. system, T = 2pisqrt(displacement / acceleration-per-unit-displacement) -- a template reused for every other oscillator in the chapter (pendulum, torsional disc, vibrating magnet) simply by substituting that system's own force or torque law. The frequency n is the number of oscillations per unit time, n = 1/T, which for a spring-block system becomes n = (1/2*pi)*sqrt(k/m): stiffer springs (larger k) give higher frequency, heavier blocks (larger m) give lower frequency. …