Simple Harmonic Motion is the gateway to every oscillation question in IB Physics HL. It underpins waves (Topic C.2–C.5), AC circuits, atomic vibrations, and even the quantum harmonic oscillator. The HL syllabus extends the SL treatment with a quantitative energy analysis ($E_T$, $E_K$, $E_P$ in terms of $\omega$, $x_0$ and $x$) and the full sinusoidal equations of motion, including a phase angle $\varphi$ set by initial conditions.
This cheatsheet condenses the full Topic C.1 syllabus — defining equation, period and frequency formulas, mass-spring and pendulum systems, energy in SHM, phase relations, and graph interpretation — into one page you can revise from. The most common HL traps (SUVAT misuse, Moon-pendulum, $E_T \propto x_0^2$) are flagged in red. Scroll to the bottom for the printable PDF download and the full Photon Academy notes library.
§1 — Conditions & Defining Equation C.1 SL + HL
Defining equation of SHM
Acceleration is proportional to displacement and always directed towards equilibrium. Here $\omega$ is the angular frequency in $\mathrm{rad\,s^{-1}}$.
Key positions
| Position | $|a|$ | $|v|$ |
|---|---|---|
| Equilibrium ($x = 0$) | 0 | maximum |
| Amplitude ($x = \pm x_0$) | maximum | 0 |
§2 — Period, Frequency & Angular Frequency C.1 SL + HL
Period–frequency relations
Mass–spring system
Springs in parallel: $k_\text{eff} = nk$ (stiffer, shorter $T$). Springs in series: $k_\text{eff} = k/n$ (softer, longer $T$). A mass between two identical springs of constant $k$ each behaves as if $k_\text{eff} = 2k$.
Simple pendulum
Valid only for small angles, $\theta_\max \lesssim 10^\circ$.
§3 — Energy in SHM C.1 SL (qualitative), HL (quantitative)
Qualitative (SL + HL)
Kinetic energy $E_K$ is maximum at equilibrium; potential energy $E_P$ is maximum at amplitude. Total energy $E_T = E_K + E_P$ is constant in the absence of damping. KE and PE exchange four times per cycle.
Quantitative (HL only)
§4 — Phase Angle & Equations of Motion C.1 HL
General sinusoidal equations (HL)
$\varphi$ is the initial phase angle (rad), set by the initial conditions at $t = 0$.
| Initial condition | $\varphi$ |
|---|---|
| Starts at $+x_0$ (cosine form) | $\varphi = \pi/2$ |
| Starts at equilibrium, moving positive | $\varphi = 0$ |
| Starts at $-x_0$ | $\varphi = -\pi/2$ |
§5 — Graphs of $x$, $v$ and $a$ C.1 SL + HL
Take $x = x_0 \cos(\omega t)$ as the canonical "starts at amplitude" case. Differentiating once gives velocity, twice gives acceleration:
| Quantity | Equation | Peak value | At $t = 0$ |
|---|---|---|---|
| Displacement $x$ | $x_0 \cos(\omega t)$ | $x_0$ | $+x_0$ (peak) |
| Velocity $v$ | $-\omega x_0 \sin(\omega t)$ | $\omega x_0$ | $0$ |
| Acceleration $a$ | $-\omega^2 x_0 \cos(\omega t)$ | $\omega^2 x_0$ | $-\omega^2 x_0$ (trough) |
Phase summary
- $x$ and $a$ are in antiphase: $\Delta\varphi = \pi$.
- $v$ leads $x$ by $\pi/2$.
- $v$ and $a$ have a phase difference of $\pi/2$.
§6 — Exam Attack Plan All sections
When you see this in the question — reach for that:
| Question trigger | Reach for |
|---|---|
| "Show that the motion is SHM" | Derive $a = -\omega^2 x$ from forces; identify $\omega^2$. |
| Mass on a spring, find $T$ | $T = 2\pi\sqrt{m/k}$. Independent of $g$. |
| Pendulum on the Moon / mountain | $T = 2\pi\sqrt{l/g}$. Smaller $g \Rightarrow$ larger $T$. |
| "Find max speed / max acceleration" | $v_\max = \omega x_0$, $a_\max = \omega^2 x_0$. |
| "Find speed at displacement $x$" | $v = \pm\omega\sqrt{x_0^2 - x^2}$ (energy method). |
| "$E_K = E_P$ at what $x$?" | $x = x_0/\sqrt{2}$ (HL). |
| "Doubling amplitude does what to energy?" | $E_T \propto x_0^2 \Rightarrow$ multiply by 4 (HL). |
| Initial conditions ($x_0$, $v_0$ at $t=0$) | $x = x_0 \sin(\omega t + \varphi)$ — solve for $\varphi$ (HL). |
| Reads off $a$–$x$ graph gradient | Gradient $= -\omega^2 \Rightarrow$ find $T = 2\pi/\omega$. |
| Phase difference between two oscillators | Compare $\varphi$; convert time lag $\Delta t$ via $\Delta\varphi = \omega \Delta t$. |
Worked Example — IB-Style HL SHM Problem
Question (HL Paper 2 style — 7 marks)
A 0.250 kg mass is attached to a horizontal spring of force constant $k = 16.0\ \mathrm{N\,m^{-1}}$ on a frictionless surface. The mass is pulled 8.0 cm from equilibrium and released from rest. Calculate (a) the angular frequency $\omega$, (b) the period $T$, (c) the maximum speed of the mass, and (d) the speed of the mass when it is 4.0 cm from equilibrium.
Solution
- Angular frequency from $\omega = \sqrt{k/m}$: $\omega = \sqrt{16.0 / 0.250} = \sqrt{64} = 8.00\ \mathrm{rad\,s^{-1}}$ (M1)(A1)
- Period from $T = 2\pi/\omega$: $T = 2\pi / 8.00 = 0.785\ \mathrm{s}$ (A1)
- Maximum speed from $v_\max = \omega x_0$ with $x_0 = 0.080$ m: $v_\max = 8.00 \times 0.080 = 0.640\ \mathrm{m\,s^{-1}}$ (M1)(A1)
- Speed at $x = 0.040$ m using $v = \omega\sqrt{x_0^2 - x^2}$: $v = 8.00 \times \sqrt{0.080^2 - 0.040^2} = 8.00 \times \sqrt{0.00480} = 0.554\ \mathrm{m\,s^{-1}}$ (M1)(A1)
- State result with units to 3 s.f. as required by IB conventions. (R1)
Examiner's note: The most common error is using SUVAT — "$v^2 = u^2 + 2as$" — to find the speed at $x = 4$ cm. Acceleration in SHM is not constant, so SUVAT is inadmissible and zero credit is awarded. The correct route is the energy formula $v = \pm\omega\sqrt{x_0^2 - x^2}$.
Common Student Questions
Why can I never use SUVAT for SHM?
Does a pendulum or a mass-spring system change period on the Moon?
What happens to the total energy if I double the amplitude?
What is the phase difference between displacement, velocity, and acceleration in SHM?
When are kinetic and potential energy equal in SHM?
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