The solution: sine and cosine

Which functions, differentiated twice, return to themselves with a sign change? Sine and cosine. This is no coincidence: they are exactly the functions that solve $\ddot x = -\omega^2 x$. The general solution of the SHM equation is

$\ev{x(t) = A\cos(\omega t + \varphi)}$

where the three parameters have distinct meanings:

• $A$ is the amplitude (m), the maximum displacement from the equilibrium position;
• $\omega$ is the angular frequency (rad/s), related to the period $T$ and the frequency $f$ by $T = \frac{2\pi}{\omega}, \qquad f = \frac{1}{T} = \frac{\omega}{2\pi}$
• $\varphi$ is the initial phase (rad), which depends on the position and velocity of the body at $t = 0$.

Intuition

$A$, $\omega$ and $\varphi$ are “the amplitude”, “how fast” and “where it starts from”. They completely fix the motion: given these three numbers, we know where the body is at every instant.

Check. Let us differentiate $x(t)$ twice and verify that it satisfies the equation:

$\begin{aligned} \dot x(t) &= -A\,\omega\,\sin(\omega t + \varphi) \\ \ddot x(t) &= -A\,\omega^2\,\cos(\omega t + \varphi) = -\omega^2\,x(t) \end{aligned}$

Bingo: the acceleration is proportional to $x$ with the opposite sign and constant $\omega^2$. The sinusoid really is the solution.

Key formula — Period of the mass-spring system

$\ev{T = 2\pi\sqrt{\frac{m}{K}}}$ Note the independence from the amplitude: $T$ does not depend on how far we pull the mass, as long as the spring stays linear. This is the isochronism of small oscillations, already observed by Galileo on the swinging lamps of Pisa Cathedral.

Why $2\pi$ appears

The $2\pi$ represents one complete turn: harmonic motion is the projection onto an axis of uniform circular motion of radius $A$. One back-and-forth oscillation corresponds to one turn through the angle $\omega t$.

Collegamenti

Argomenti: Oscillations and harmonic motion Concetti: Simple harmonic motion · Period and frequency Oggetti: Spring

Esercizi collegati: Problem — Period and frequency of a mass-spring system · Problem — Natural frequency of a skyscraper · Problem — Ranking springs and masses