Mass-spring without friction

Second-order differential equations describe an enormous variety of physical phenomena. The first key example is the ideal mass-spring.

Example 1 — Mass-spring without friction

A mass $m$ attached to a spring of constant $k$ obeys Newton’s second law $m\ddot x = -kx$, that is: $\ddot x + \omega_0^2 x = 0, \qquad \omega_0 = \sqrt{k/m}.$ Characteristic equation $\lambda^2 + \omega_0^2 = 0$, roots $\lambda = \pm i\omega_0$. General solution: $x(t) = A\cos(\omega_0 t) + B\sin(\omega_0 t) = C\cos(\omega_0 t + \varphi).$ With initial conditions $x(0)=x_0$, $\dot x(0)=0$: $A = x_0$, $B=0$, hence $x(t) = x_0\cos(\omega_0 t)$.

The period is $T = 2\pi/\omega_0 = 2\pi\sqrt{m/k}$: larger mass $\Rightarrow$ longer period; stiffer spring $\Rightarrow$ shorter period. Numerically, with $m=0{,}5$ kg and $k=200$ N/m we have $\omega_0 = 20$ rad/s and $T \approx 0{,}31$ s.

Links

Topics: Differential equations
Concepts: Initial condition · Second-order linear ODE · Characteristic equation · Harmonic oscillator
Functions: Cosine
Skills: Modelling · Solving equations