- $F$ restoring force on the mass (N)
- $k$ spring constant — force per unit displacement (N m⁻¹)
- $x$ displacement from equilibrium (m)
The minus sign is the entire physical content: the force always points back toward equilibrium, regardless of which side $x$ is displaced to. This — force proportional to displacement, directed opposite to it — is the defining property of every simple harmonic oscillator, not just springs.
- $\omega$ angular frequency of oscillation (rad s⁻¹)
- $x$ displacement, as a function of time
Every equation in this chapter is either deriving this equation for a new physical system, or solving it. If you can recognize a system reduces to this form, you already know its entire behavior.
- $k$ spring constant (N m⁻¹)
- $m$ mass (kg)
Key property: $\omega$ depends only on $k$ and $m$ — the physical properties of the oscillator — never on amplitude or how the motion was started. This is why a pendulum clock keeps accurate time regardless of how far you initially pull it.
- $A$ amplitude — maximum displacement (m)
- $\omega$ angular frequency, fixed by the oscillator's physical properties
- $\phi$ phase angle (rad) — set by initial conditions
Equation 1.6 is a second-order differential equation, so its general solution always has exactly two free constants. Here they're $A$ and $\phi$; equivalently it can be written $x = a\cos\omega t + b\sin\omega t$ with $a = A\cos\phi$, $b=-A\sin\phi$ (Eq. 1.15–1.17) — same solution, different-looking constants.
Straightforward differentiation of $x=A\cos\omega t$ with respect to time, once and twice. Worth noticing directly from the graphs: velocity is zero exactly when displacement is at its extremes (the turning points), and maximum exactly when displacement passes through zero — a quarter-period phase shift between the two.
- $K$ kinetic energy, $\tfrac12mv^2$
- $U$ potential energy stored in the spring, $\tfrac12kx^2$ (from integrating $F=-kx$, Eq. 1.18)
Start from $m\,\mathrm dv/\mathrm dt = -kx$, multiply through by $\mathrm dx = v\,\mathrm dt$ to get $mv\,\mathrm dv = -kx\,\mathrm dx$, then integrate both sides — the left becomes $\tfrac12mv^2$, the right $-\tfrac12kx^2$, and together they sum to a constant: the total energy $E$.
Substituting the general solution into $K+U$, the time-dependent $\sin^2$ and $\cos^2$ terms combine to exactly $1$ (Eq. 1.21–1.23), leaving a constant — confirming energy conservation directly, and showing energy scales with the square of the amplitude. Double the amplitude, quadruple the energy.
Any smooth potential well $U(x)$ can be Taylor-expanded about its minimum (Eq. 1.25). The constant term is irrelevant (potential energy has no absolute zero), and the first-derivative term vanishes exactly because $x=0$ is a minimum. What survives, to leading order, is a term proportional to $x^2$ — i.e. a parabolic well, which always produces a linear restoring force. This is the actual reason SHM is universal: any system in stable equilibrium behaves like a harmonic oscillator for sufficiently small displacements, whether it's a pendulum, a diatomic molecule, or a marble in a bowl.
- $\theta$ angular displacement from vertical (rad)
- $l$ length of the pendulum (m)
- $g$ acceleration due to gravity
The exact equation has $\sin\theta$, not $\theta$, so it isn't literally SHM. Expanding $\sin\theta = \theta - \theta^3/3! + \cdots$ (Eq. 1.30) and keeping only the first term — valid once $\theta$ is small enough that higher powers are negligible — recovers the SHM form exactly.
Result: $\omega = \sqrt{g/l}$, $T = 2\pi\sqrt{l/g}$ (Eq. 1.32). Note $\omega$ depends on $g$ and $l$ only — not mass, not amplitude.
- $\alpha$ the "inertia" constant (mass for a spring, $\alpha=m$; moment of inertia for a rotating system)
- $\beta$ the "stiffness" constant (spring constant $k$; or $mg/l$ for a pendulum)
Both the mass-on-spring energy ($\tfrac12mv^2+\tfrac12kx^2$) and the pendulum energy ($\tfrac12mv^2+\tfrac12\frac{mg}{l}x^2$) are special cases of this one form. Differentiating it with respect to time always reproduces the SHM equation with $\omega=\sqrt{\beta/\alpha}$ — meaning you never need to redo this derivation for a new system: just identify its $\alpha$ and $\beta$ and read off $\omega$ directly.
- $q$ charge on the capacitor (C)
- $L$ inductance (H)
- $C$ capacitance (F)
Kirchhoff's voltage law, $V_C + V_L = 0$, with $V_C = q/C$ and $V_L = L\,\mathrm dI/\mathrm dt = L\,\mathrm d^2q/\mathrm dt^2$ (since $I=\mathrm dq/\mathrm dt$). Same mathematical form as Eq. 1.6, with $\omega = \sqrt{1/LC}$.
The correspondence (Eq. 1.57–1.58): $q \leftrightarrow x$, $L \leftrightarrow m$, $1/C \leftrightarrow k$. Every mechanical SHM result translates directly — this is why electrical circuits can be built as analogue simulations of mechanical systems.
- $b$ damping constant — force per unit velocity (N s m⁻¹)
- $v$ velocity of the mass
This linear-in-$v$ form is an approximation (valid for velocities that aren't too large), but it's the one that keeps the equation of motion linear and solvable — a damping force proportional to $v^2$ would be far harder to handle analytically.
- $\omega_0^2$ $=k/m$ — the natural frequency: what the system would oscillate at with zero damping
- $\gamma$ $=b/m$ — the damping parameter (Eq. 2.4)
Newton's second law with both forces present: $ma = -kx - bv$ (Eq. 2.3a), rearranged and divided through by $m$.
Everything else in this chapter is one of three cases determined by comparing $\gamma^2/4$ to $\omega_0^2$.
$\gamma^2/4 < \omega_0^2$ — damping is weak enough that $\omega$ stays real, so the system still oscillates, just at a slightly lower frequency than $\omega_0$, with an amplitude that decays as $\exp(-\gamma t/2)$.
- $A_0$ initial amplitude
- $\exp(-\gamma t/2)$ the decaying envelope — not the oscillation itself, just the ceiling it decays within
- $\cos(\omega t+\phi)$ the oscillation inside that envelope, at the (slightly reduced) angular frequency $\omega$
- $A_n$ amplitude of the $n$-th peak; $A_{n+1}$ the next one, one period $T$ later
Since successive peaks always shrink by the same fractional amount, measuring any two consecutive peaks from a graph gives you $\gamma$ directly — a standard way to extract the damping parameter from experimental data.
$\gamma^2/4 > \omega_0^2$ — damping dominates so strongly that the system just relaxes back to equilibrium, sluggishly, with two different exponential decay rates and no crossing back and forth.
Both exponents are real and negative (since $\sqrt{\gamma^2/4-\omega_0^2} < \gamma/2$ always), so both terms are pure exponential decays — nothing in the solution oscillates.
$\gamma^2/4 = \omega_0^2$ exactly — the boundary case between light and heavy damping.
Of all three damping regimes, this is the one that returns the system to equilibrium in the shortest possible time without ever oscillating past it. That's why it's the design target for shock absorbers, door closers, and analogue meter needles — anything that needs to settle quickly without bouncing.
- $E_0$ total energy at $t=0$
- $\tau = 1/\gamma$ the decay time (or time constant) — the time for energy to fall by a factor of $e$
Substitute the light-damping solution into $E=\tfrac12mv^2+\tfrac12kx^2$; for very light damping the oscillating $\sin^2$ and $\cos^2$ terms again combine to $1$ (as in Ch. 1), leaving a clean exponential decay envelope.
$Q$ is a dimensionless "figure of merit" that works for any kind of oscillator — mechanical, electrical, or otherwise — regardless of what units its damping is measured in. Comparing the decay time $\tau$ (how long energy takes to dissipate) to the period $T$ (how long one oscillation takes) is the essential idea: many oscillations before dying out means high $Q$.
- high $Q$ many oscillations before the energy is significantly reduced (quartz crystal: $Q\sim10^6$)
- low $Q$ only a few oscillations, or none (paperweight on a rubber band: $Q\sim10$)
Same physics as Eq. 2.5, with $\gamma$ replaced by $\omega_0/Q$ — useful because $Q$ is often the quantity actually measured or quoted for a real oscillator. Note that even a fairly low $Q=5$ only shifts $\omega$ from $\omega_0$ by about 0.5%, confirming that $\omega\approx\omega_0$ is an excellent approximation whenever damping is light.
- $q\leftrightarrow x$, $L\leftrightarrow m$, $R\leftrightarrow b$, $1/C\leftrightarrow k$
Since the differential equations are identical in form, every mechanical damping result carries over directly: light/heavy/critical damping, energy decay, and $Q$ all apply unchanged with these substitutions.
Direct substitution of the mechanical-to-electrical correspondence above into Eq. 2.6 and Eq. 2.21. Larger resistance $R$ means more damping (smaller $Q$), exactly as a larger $b$ would mechanically.