Oscillations

Five topics on motion that repeats. Simple harmonic motion is any motion obeying $a = -\omega^2 x$, where the restoring force pulls the object back toward equilibrium in proportion to displacement; the period $T = 2\pi\sqrt{m/k}$ for a spring and $T = 2\pi\sqrt{L/g}$ for a small-amplitude pendulum depends on the system, not the amplitude. The motion is captured by $x(t) = A\cos(\omega t + \phi)$ with velocity $v = -A\omega\sin(\omega t + \phi)$ and acceleration $a = -A\omega^2\cos(\omega t + \phi)$; energy sloshes between $U = \tfrac{1}{2}kx^2$ and kinetic energy while the total $E = \tfrac{1}{2}kA^2$ stays fixed; and physical pendulums extend the idea to extended bodies through $T = 2\pi\sqrt{I/(mgd)}$.

Topics

7.1

Defining Simple Harmonic Motion (SHM)

Frequency and Period of SHM

Representing and Analyzing SHM

Energy of Simple Harmonic Oscillators

Simple and Physical Pendulums

Lesson

Diag

locked

Equations For every problem in this unit

Position

$x(t) = A\cos(\omega t + \phi)$

Velocity

$v(t) = -A\omega\sin(\omega t + \phi)$

Acceleration

$a(t) = -A\omega^2\cos(\omega t + \phi)$

SHM condition

$a = -\omega^2 x$

Angular frequency

$\omega = \dfrac{2\pi}{T} = 2\pi f$

Spring-mass period

$T = 2\pi\sqrt{m/k}$

Simple pendulum period

$T = 2\pi\sqrt{L/g}$

Physical pendulum period

$T = 2\pi\sqrt{I/(mgd)}$

Total energy

$E = \tfrac{1}{2}kA^2$

Potential energy

$U = \tfrac{1}{2}kx^2$

Maximum speed

$v_{max} = A\omega$

Maximum acceleration

$a_{max} = A\omega^2$

Unit 7 tools

Challenge bank

1 / 50

50 open-ended problems.

Read the question, work it out, then flip the card to compare your reasoning to the worked solution. Mark each card so you can return to the ones that still bite.

Question

Tap card to reveal explanation

Worked solution

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Cumulative assessment

Test the unit.

Twenty mixed items pulled from across all 5 topics, identifying which misconceptions still bite when the question does not tell you which topic it came from.