Ch 1 · 17 Oscillations · A Level Physics 9702
Topic 1 of 1 in Oscillations — A Level Physics 9702 — 67 lessons.
Introduction
A pendulum, a mass on a spring, a guitar string, the atoms in a crystal and a building swaying in an earthquake all oscillate. Many of them follow the same mathematical pattern, simple harmonic motion, and the same idea of resonance explains both musical instruments and collapsing bridges. (Syllabus topic 17: sections 17.1–17.3.)
Key ideas — 17.1 (a) Describing oscillations
T = 1/f = 2π/ω
🧑🏫 Teacher's explanation
Imagine a child on a swing. When they are sitting perfectly still at the bottom, their displacement is zero. As they swing forward, they move away from that center point. The furthest point they reach before falling back is the amplitude. Simple Harmonic Motion (SHM) is a special kind of repetitive movement where the 'pull' back to the center gets stronger the further away you go.
📖 Definitions
🔍 Worked example — Easy
A mass on a spring oscillates with a frequency of 5.0 Hz. Calculate the angular frequency ω.
🔍 Worked example — Standard
An object undergoes SHM with an amplitude of 0.20 m and an angular frequency of 4.0 rad s^-1. Calculate the maximum acceleration of the object.
⚠️ Common errors
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🎯 Exam question
Key ideas — 17.1 (b) Simple harmonic motion
Simple harmonic motion (SHM) occurs when the acceleration is proportional to the displacement from a fixed point and in the opposite direction (always towards the equilibrium position):
🧑🏫 Teacher's explanation
Imagine a mass attached to a spring on a frictionless surface. If you pull it and let go, it doesn't just move back and forth at a constant speed; it slows down as it reaches the furthest points and rushes through the middle. This 'restoring' behavior is the essence of Simple Harmonic Motion (SHM).
📖 Definitions
🔍 Worked example — Easy
A mass on a spring oscillates with an amplitude of 0.05 m and a frequency of 2.0 Hz. Calculate the angular frequency ω.
🔍 Worked example — Standard
An object undergoes SHM with an angular frequency ω = 5.0 rad s^-1 and amplitude x_0 = 0.20 m. Calculate the maximum velocity v_max.
🔍 Worked example — Exam-style
A 0.50 kg mass is oscillating with amplitude x_0 = 0.10 m and angular frequency ω = 4.0 rad s^-1. Calculate the total energy of the system.
⚠️ Common errors
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🎯 Exam question
Key ideas — 17.1 (c) Graphs
🧑🏫 Teacher's explanation
Imagine a person on a swing. The displacement is how far they are from the center. When they are at the highest point, they stop for a split second (velocity is zero), but they are about to turn back. This 'turning point' is why the velocity graph looks like a cosine wave—it's shifted relative to the displacement.
📖 Definitions
🔍 Worked example — Easy
A particle undergoes SHM with displacement x = 0.20 sin(4 rad exts^-1t). Determine the maximum velocity v_0.
🔍 Worked example — Standard
An object oscillates with an amplitude of 0.05 m and a frequency of 5.0 Hz. Calculate the maximum acceleration a_max.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
(a) State the amplitude of the oscillation. [1] (b) Calculate the maximum velocity of the mass. [2] (c) Calculate the maximum acceleration of the mass. [2] [5 marks]
Key ideas — 17.2 Energy in SHM
During SHM, energy is continually exchanged between kinetic and potential energy (elastic PE for a spring, gravitational PE for a pendulum):
🧑🏫 Teacher's explanation
Imagine a child on a playground swing. When they are at the very bottom of the arc (the equilibrium position), they are moving at their fastest; they have lots of 'oomph' or kinetic energy, but they aren't high up, so they have very little potential energy. As they swing higher and higher, they slow down. That 'oomph' is being converted into height (potential energy).
📖 Definitions
🔍 Worked example — Easy
A mass of 0.50 kg is attached to a spring and undergoes SHM with an angular frequency ω = 4.0 rad s^-1. Calculate the total energy of the system if the amplitude is 0.20 m.
🔍 Worked example — Standard
An object in SHM has a total energy of 10.0 J. When its displacement is x = 0.50 m, its potential energy is 4.0 J. Calculate the kinetic energy at this displacement.
🔍 Worked example — Exam-style
A particle of mass m oscillates with angular frequency ω. If the amplitude is doubled, by what factor does the maximum kinetic energy increase?
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — 17.3 (a) Damping
Damping is the loss of energy from an oscillating system caused by a resistive force (friction, air resistance, viscous drag). The amplitude decreases with time.
🧑🏫 Teacher's explanation
Imagine you are pushing a child on a swing. If there were no air resistance or friction in the chains, the child would swing back and forth forever with the same height (amplitude). However, in the real world, air resistance and friction act as resistive forces that constantly steal energy from the system. This process is called damping.
📖 Definitions
🔍 Worked example — Easy
A mass-spring system is subject to light damping. If the initial amplitude is 0.10 m, and after 2.0 s the amplitude has reduced to 0.05 m, calculate the ratio of the amplitude at t=0 to the amplitude at t=2.0\ ext s.
🔍 Worked example — Standard
A car suspension system is designed to be critically damped. If the system is displaced by 0.050 m, describe the motion of the car body and state the primary goal of this damping type.
🔍 Worked example — Exam-style
An oscillating system has a total energy E = 1/2mω^2x_0^2. If damping causes the amplitude x_0 to halve, by what factor does the total energy decrease?
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — 17.3 (b) Forced oscillations and resonance
Every oscillating system has a natural frequency f_0, at which it oscillates when displaced and released.
🧑🏫 Teacher's explanation
Imagine you are pushing a friend on a swing. If you push them at random times, they won't go very high. But if you time your pushes perfectly to match the natural rhythm of the swing, they will go higher and higher with very little effort. This 'perfect timing' is resonance.
📖 Definitions
🔍 Worked example — Easy
A mass-spring system has a natural frequency of f_0 = 5.0 Hz. If a driving force is applied at a frequency of f_d = 5.0 Hz, describe the state of the system's amplitude.
🔍 Worked example — Standard
A system undergoes forced oscillations. The amplitude A as a function of driving frequency f is given by a curve that peaks at f = 12 Hz. If the damping is increased, how will the peak amplitude and the width of the peak change?
🔍 Worked example — Exam-style
A pendulum has a natural frequency of 2.0 Hz. It is driven by a motor at 2.5 Hz. If the damping is 'heavy', calculate the qualitative change in the amplitude-frequency graph compared to a 'lightly damped' system.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Worked example
Example 1. A mass on a spring oscillates in SHM with amplitude 0.050 m and period 0.40 s. Find (a) ω, (b) the maximum speed, (c) the maximum acceleration, (d) the speed at x = 0.030 m.
🔍 Worked example — Easy
A mass on a spring oscillates in SHM with amplitude x_0 = 0.050 m and period T = 0.40 s. Find (a) the angular frequency ω, (b) the maximum speed v_max, and (c) the maximum acceleration a_max.
🔍 Worked example — Standard
A mass of 0.20 kg oscillates with an amplitude of 0.050 m and an angular frequency of 15.7 rad s^-1. Calculate the total energy E of the system.
🔍 Worked example — Exam-style
An object in SHM has an amplitude of 0.050 m and an angular frequency of 15.7 rad s^-1. Calculate its speed when its displacement is 0.030 m.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) Motion in which the acceleration is proportional to the displacement from a fixed point and always directed towards that point (opposite to the displacement). (2) T = 2.0 s; ω = π = 3.14 rad s⁻¹. (3) A straight line through the origin with negative gradient; gradient = -ω^2. (4) v = 10√(0.040^2 - 0.020^2) = 0.35 m s⁻¹. (5) It quadruples (E ∝ x_0^2). (6) Light damping: the system oscillates with gradually decreasing amplitude; critical damping: it returns to equilibrium in the shortest time without oscillating.