Springs In Combination

Springs combined in parallel share the load so their stiffnesses add, a result following from the force analysis in NCERT Class 11, Chapter 14 (Oscillations); the effective constant becomes $k_{eff}=k+k=2k$. Since the period of a spring-mass system is $T=2\pi \sqrt{m/k_{eff}}$, a larger effective stiffness gives a shorter period. Comparing the parallel case with a single spring, $\frac{T_{parallel}}{T_{single}}=\sqrt{\frac{k}{2k}}=\frac{1}{\sqrt{2}}$. Thus the period decreases by a factor of $\sqrt{2}$. The option that it increases by $\sqrt{2}$ describes a series combination, where the effective constant is halved, not doubled. The option that it stays the same ignores that parallel springs stiffen the system. The option that it halves would require the effective constant to quadruple, which is not the case for two equal springs. The physical reason the constants add in parallel is that both springs stretch by the same displacement $x$, so each contributes a restoring force $kx$ and the total is $2kx$, equivalent to a single spring of constant $2k$. Contrast this with a series arrangement, where the same force stretches both springs and the extensions add, giving a softer effective constant $k/2$ and a longer period. A sanity check: stiffer support should make oscillations faster, so a smaller period is physically expected, and the factor $1/\sqrt{2}\approx 0.71$ is a modest, reasonable reduction rather than an extreme change.