IBDP Physics SL Cheat sheet - C.1 Simple harmonic motion | IBDP Physics Cheatsheets

What SHM is

Simple harmonic motion (SHM) is oscillation about an equilibrium position where the restoring acceleration is directly proportional to displacement and always directed toward equilibrium.
Defining equation: $a=-\omega^2x$
The minus sign is crucial: it shows acceleration is opposite to displacement, so the motion is restoring, not runaway.
SHM occurs when the resultant force is restoring and behaves like $F\propto -x$ .
Typical IB examples: mass–spring system and simple pendulum (for small angles).
At equilibrium: $x=0$ , acceleration = 0, speed is maximum.
At extremes: $x=\pm A$ , speed = 0, magnitude of acceleration is maximum.
Amplitude $A$ = maximum displacement from equilibrium.
Period $T$ = time for one complete oscillation.
Frequency $f$ = number of oscillations per second.
Angular frequency $\omega$ is measured in rad s $^{-1}$ .

This diagram shows a mass–spring oscillator at several positions in one cycle, with the direction of restoring force at each point. It is useful for linking displacement, velocity, and force qualitatively. Source

Core quantities and relationships

$T=\dfrac{1}{f}=\dfrac{2\pi}{\omega}$
Therefore $f=\dfrac{1}{T}$ and $\omega=2\pi f$ .
In calculations, keep units consistent: $T$ in s, $f$ in Hz, $\omega$ in rad s $^{-1}$ , $x$ and $A$ in m.
For SHM, period does not depend on displacement at a particular instant.
For a given oscillator, larger $\omega$ means shorter period.
In exam questions, identify whether the system is asking for time period, frequency, or angular frequency before substituting.

This graph shows the classic sinusoidal displacement–time shape of SHM with amplitude and period marked. It is ideal for recognizing how $A$ and $T$ are read directly from a motion graph. Source

Mass–spring systems

A mass attached to a spring performs SHM when the restoring force follows Hooke’s law: $F=-kx$ .
Comparing with $F=ma$ gives $a=-\dfrac{k}{m}x$ , so $\omega^2=\dfrac{k}{m}$ .
Time period of a mass–spring system: $T=2\pi\sqrt{\dfrac{m}{k}}$
Larger mass $m$ gives larger period.
Larger spring constant $k$ gives smaller period.
For IB exam questions, remember $T$ is independent of amplitude for ideal SHM.
If asked whether motion is SHM, check whether the resultant force/acceleration is proportional to displacement and opposite in direction.

Simple pendulum

A simple pendulum performs approximate SHM only for small angular displacements.
Time period of a simple pendulum: $T=2\pi\sqrt{\dfrac{l}{g}}$
Larger length $l$ gives larger period.
Larger gravitational field strength $g$ gives smaller period.
The mass of the bob does not affect the period.
This formula applies to a simple pendulum and to small-angle oscillations only.

This diagram shows the geometry of a simple pendulum, including displacement from the vertical and the change in height. It helps explain why a pendulum has a restoring effect and why the small-angle model is used for SHM. Source

Displacement, velocity and acceleration

In SHM, displacement and acceleration are always in opposite directions.
The magnitude of acceleration increases as the particle moves further from equilibrium.
Speed/velocity magnitude is maximum at equilibrium and zero at the extremes.
From the defining equation, when $x=0$ , then $a=0$ .
When $x=\pm A$ , the magnitude of acceleration is maximum: $a_{\max}=\omega^2A$ .
A common exam trap: maximum acceleration does not occur where speed is maximum.
Graphically:
$x$ – $t$ is sinusoidal.
$v$ – $t$ is sinusoidal and $\frac{\pi}{2}$ out of phase with displacement.
$a$ – $t$ is sinusoidal and in antiphase with displacement.
For gradient reasoning: gradient of $x$ – $t$ gives velocity; gradient of $v$ – $t$ gives acceleration.

This image shows displacement, velocity, and acceleration curves for SHM on the same set of axes. It is especially useful for seeing the phase relationships and identifying where each quantity is zero or maximum. Source

HL only: phase angle and equations

A particle in SHM can be described by a phase angle $\phi$ .
Use radians for phase calculations.
Displacement equation: $x=x_0\sin(\omega t+\phi)$
Velocity equation: $v=\omega x_0\cos(\omega t+\phi)$
Speed-displacement relation: $v=\pm\omega\sqrt{x_0^2-x^2}$
Here $x_0$ is the amplitude.
The sign of $v$ depends on the direction of motion.
At maximum displacement $x=\pm x_0$ , velocity is zero.
At equilibrium $x=0$ , the speed is maximum: $v_{\max}=\omega x_0$ .
Phase lets you determine the oscillator’s position within a cycle.

This phase chart links the oscillator’s position in the cycle to its changing motion variables. It is helpful for HL students interpreting phase angle and tracking where the oscillator is during one full oscillation. Source

Energy changes in SHM

In SHM, total energy remains constant if there is no damping.
Energy continually transfers between kinetic energy and potential energy.
At the extremes: potential energy is maximum, kinetic energy is zero.
At equilibrium: kinetic energy is maximum, potential energy is minimum.
The oscillator never has maximum kinetic energy and maximum potential energy at the same point.
SL requirement: describe these energy changes qualitatively over one cycle.
HL quantitative energy equations:
$E_T=\dfrac{1}{2}m\omega^2x_0^2$
$E_p=\dfrac{1}{2}m\omega^2x^2$
Therefore $E_k=E_T-E_p=\dfrac{1}{2}m\omega^2(x_0^2-x^2)$
As $|x|$ increases, $E_p$ increases and $E_k$ decreases.

This graph shows how kinetic energy, elastic potential energy, and total energy vary in undamped SHM. It is excellent for identifying why the total energy stays constant while the other two exchange throughout the cycle. Source

Checklist: can you do this?

State and interpret the defining condition for SHM: $a=-\omega^2x$ .
Calculate and convert between $T$ , $f$ , and $\omega$ .
Use $T=2\pi\sqrt{\dfrac{m}{k}}$ and $T=2\pi\sqrt{\dfrac{l}{g}}$ correctly.
Identify where speed, acceleration, kinetic energy, and potential energy are maximum or zero.
HL: solve for $x$ , $v$ , phase, and energy using the SHM equations.

Written by:

Dr Shubhi Khandelwal

Qualified Dentist and Expert Science Educator

Shubhi is a seasoned educational specialist with a sharp focus on IB, A-level, GCSE, AP, and MCAT sciences. With 6+ years of expertise, she excels in advanced curriculum guidance and creating precise educational resources, ensuring expert instruction and deep student comprehension of complex science concepts.