Academy — Sanjib Sikder

How spring constant k and mass m determine the time period — and what changes when springs are combined.

The spring-mass system is the prototype of SHM. Every result here applies — with appropriate substitution — to pendulums, LC circuits, and molecular vibrations.

The time period

Newton's second law + Hooke's law:

$m\ddot{x} = -kx \implies \omega = \sqrt{\frac{k}{m}}, \quad \boxed{T = 2\pi\sqrt{\frac{m}{k}}}$

Heavier mass → slower oscillation

$T \propto \sqrt{m}$ . Quadrupling mass doubles the period.

Light mass ( $m = 1$ kg):

SpringMass amplitude 60 k 100 m 1
Show spring block wall graph-x readouts
Hide energy displacement-arrow velocity-arrow acceleration-arrow equilibrium

Heavy mass ( $m = 4$ kg, same spring):

SpringMass amplitude 60 k 100 m 4
Show spring block wall graph-x readouts
Hide energy displacement-arrow velocity-arrow acceleration-arrow equilibrium

Watch the x(t) graph — cycles are twice as wide. T displayed in the readout confirms it.

Stiffer spring → faster oscillation

$T \propto 1/\sqrt{k}$ . Quadrupling stiffness halves the period.

Soft spring ( $k = 50$ N/m):

SpringMass amplitude 60 k 50 m 1
Show spring block wall graph-x readouts
Hide energy displacement-arrow velocity-arrow acceleration-arrow equilibrium

Stiff spring ( $k = 200$ N/m, same mass):

SpringMass amplitude 60 k 200 m 1
Show spring block wall graph-x readouts
Hide energy displacement-arrow velocity-arrow acceleration-arrow equilibrium

Springs in series

Two springs end-to-end. Same force stretches both — displacements add:

$\frac{1}{k_{eff}} = \frac{1}{k_1} + \frac{1}{k_2} \implies k_{eff} = \frac{k_1 k_2}{k_1 + k_2}$

Effective stiffness is less than either spring. Period is longer.

Springs in parallel

Two springs side by side. Same displacement — forces add:

$k_{eff} = k_1 + k_2$

Effective stiffness is greater. Period is shorter.

Cutting a spring

A spring of constant $k$ cut to $\frac{1}{n}$ of its length has constant $nk$ .

Fewer coils → each coil must stretch more per unit force → stiffer.