Waves and Optics: practice problems and exam questions

Oscillations and waves: wave speed, sound and decibels, the Doppler effect, standing waves on strings and in pipes, refraction and total internal reflection, lenses and mirrors, interference and diffraction. Most exam questions need one key relation used carefully with the right units and signs.

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Cram pack (PDF)Formula sheets, method steps and practice problems with full solutions, ready to print.

Topics in this course

  1. Springs & pendulumsmass on a spring · pendulum · speed, acceleration and energycalculations
  2. Waves, sound & opticswave speed, sound level and refraction · standing waves, lenses and Doppler · interference, gratings, mirrors and fibrescalculations

Exam tips for Waves and Optics

  • Measure every angle from the normal, never from the surface.
  • Write the sign convention next to every lens or mirror problem: f < 0 for diverging lenses and convex mirrors, dᵢ < 0 for virtual images.
  • Convert nm, mm and cm to metres before you use an interference formula.

Worked examples

Springs & pendulumsEasy
3.2 kgk = 210 N/m

A 3.2 kg mass on a spring with stiffness 210 N/m oscillates on a frictionless surface. Find the period T.

m = 3.2 kgk = 210 N/m

Find T (s)

Show hints
  1. The period of a mass–spring system does not depend on the amplitude.
  2. T = 2π·√(m/k).
  3. √(m/k) = 0.1234 s.
Show the answer and the step-by-step solution

Simple harmonic motion: a mass m on a spring of stiffness k oscillates with angular frequency ω = √(k/m) and period T = 2π/ω = 2π·√(m/k). A simple pendulum of length L has T = 2π·√(L/g). With amplitude A, the largest speed is A·ω, the largest acceleration A·ω² and the energy ½·k·A².

Period

T = 2π·√(3.2/210) = 0.7756 s

Answer

T = 0.7756 s
Waves, sound & opticsMedium
L = ?n = 3

A pipe open at both ends must have its harmonic number 3 at 686 Hz. The speed of sound is 343 m/s. Find the length L of the pipe.

v = 343 m/sn = 3f3 = 686 Hz

Find L (m)

Show hints
  1. f = n·v/(2L).
  2. Solve for L.
  3. λ = v/f = 0.5 m.
Show the answer and the step-by-step solution

Standing waves and the Doppler effect: a string fixed at both ends, or a pipe open at both ends, fits whole half-wavelengths: L = nλ/2, fn = n·v/(2L), n = 1, 2, 3, … A pipe closed at one end has a node at the closed end and an antinode at the open end: L = nλ/4 with odd n only, fn = n·v/(4L). When source or observer move along the line between them, f′ = f·(v ± vo)/(v ∓ vs): use the upper signs for motion towards each other. Two waves of nearly equal frequency beat at |f₁ − f₂|.

Wavelength

λ = v/f = 343/686 = 0.5 m

Length

L = nλ/2 = 3·0.5/2 = 0.75 m

Answer

L = 0.75 m

How practising works

  • “Practise this course” mixes problems from all its topics, like a real exam.
  • Every problem has new values, so you learn the method instead of an answer.
  • Stuck? Open a small hint first, and the full explanation only when you need it.
  • Mock exams are timed and have no hints, and you get your score and a full review at the end.

Calculators for this course

Pendulum and spring period calculatorSpring constant calculator (Hooke’s law)

Related courses

Physics 1: MechanicsAP Physics 1 practice problemsDynamicsPhysics for EngineersPhysics 2: Electricity and Magnetism
Practise this course for freePractise this courseTake a mock exam5 free problems every day. No card needed.