PHY131H1/Unit 9

Unit 9 · Introduction to Physics I

Oscillations

One condition — a restoring force proportional to displacement — produces the same motion everywhere it appears: a mass on a spring, a pendulum, an atom in a crystal, a bridge in the wind. Recognise that condition and the answer is already known.

By the end you can

• State the condition that produces simple harmonic motion.
• Compute period and frequency for springs and pendulums.
• Explain why amplitude does not affect the period.
• Describe how energy moves between kinetic and potential.

Why it matters & where it comes from

Why it exists

Near any stable equilibrium, the restoring force is approximately proportional to displacement. That makes simple harmonic motion the universal small-vibration behaviour of essentially every physical system.

In the real world

Quartz watches keep time by a crystal oscillating $32{,}768$ times a second. Car suspensions are damped oscillators tuned to absorb bumps without bouncing. Buildings in earthquake zones are designed around their natural frequency.

Story & open questions

Galileo noticed a swinging cathedral lamp kept time against his pulse regardless of how far it swung. That observation — period independent of amplitude — led to the pendulum clock and to the first accurate timekeeping in history.

The condition for SHM

Definition

Simple harmonic motion occurs whenever the restoring force is proportional to the displacement and directed back toward equilibrium:

$F = -kx$

The minus sign is essential: it says the force always points back. With $F = ma$ this gives $a = -(k/m)x$, so the acceleration is proportional to displacement and opposite in sign.

Rule The resulting motion

$x(t) = A\cos(\omega t + \phi)$, where $A$ is the amplitude and $\phi$ sets the starting point.

$\omega = \sqrt{k/m}$ is the angular frequency, in rad/s.

Period $T = 2\pi/\omega = 2\pi\sqrt{m/k}$, and frequency $f = 1/T$.

Example Where the acceleration is largest

At maximum displacement $x = \pm A$: speed is zero, acceleration is maximum. All the energy is potential.

At equilibrium $x = 0$: speed is maximum ($v_{\max} = A\omega$), acceleration is zero. All the energy is kinetic.

This is the opposite of what many expect. The object moves fastest where the force on it is zero, exactly as at the top of a throw in reverse.

Why amplitude does not matter

Rule Isochronism

The period $T = 2\pi\sqrt{m/k}$ contains no amplitude. Pull the mass twice as far and it takes exactly as long to return.

The reason is that doubling the displacement doubles the restoring force, so the object covers twice the distance at twice the acceleration — and the two effects cancel exactly. This only works because the force is linear in $x$.

Example Mass on a spring

A $0.5$ kg mass on a spring with $k = 200$ N/m.

$\omega = \sqrt{200/0.5} = 20$ rad/s.

$T = 2\pi/20 = 0.314$ s, so $f = 3.18$ Hz.

Pulled $0.1$ m and released, $v_{\max} = A\omega = 2$ m/s. Doubling the pull doubles the maximum speed but leaves the period unchanged.

Rule The simple pendulum

For small swings, $T = 2\pi\sqrt{L/g}$ — the form Unit 1 derived by dimensional analysis, now with its factor of $2\pi$.

The mass does not appear. A heavy and a light pendulum of the same length keep the same time.

The restoring force is $-mg\sin\theta$, which is only proportional to $\theta$ when $\theta$ is small enough that $\sin\theta \approx \theta$. Beyond about $15$ degrees the approximation fails and the period lengthens.

Isochronism is an approximation for pendulums

A spring is exactly isochronous because $F = -kx$ is exactly linear. A pendulum only approximately so, because $\sin\theta$ is only approximately $\theta$. Galileo's cathedral lamp was swinging through a small angle, which is why his observation held.

Energy in oscillation

Rule Total energy is constant

With no damping, $E = \tfrac12kA^2$ throughout, exchanging between forms:

$U = \tfrac12kx^2$, maximum at the extremes.

$K = \tfrac12mv^2$, maximum at the centre.

$\tfrac12kx^2 + \tfrac12mv^2 = \tfrac12kA^2$ at every instant, which gives $v = \omega\sqrt{A^2 - x^2}$ without needing time.

Energy depends on $A^2$, so doubling the amplitude quadruples the energy.

Example Speed at half amplitude

For the spring above with $A = 0.1$ m, find the speed at $x = 0.05$ m.

$v = 20\sqrt{0.01 - 0.0025} = 20 \times 0.0866 = 1.73$ m/s.

That is $87$ per cent of the maximum speed, even though it is halfway out. The speed falls off slowly near the centre and rapidly near the ends.

Definition

Damping is a resistive force, usually proportional to velocity, that removes energy. Three regimes:

Underdamped: oscillates with amplitude decaying over time. A guitar string.

Critically damped: returns to equilibrium as fast as possible without overshooting. Car suspension and door closers.

Overdamped: returns slowly, no oscillation.

Example Resonance

Driving an oscillator at its natural frequency pumps energy in efficiently, and the amplitude grows large.

Pushing a swing at the right moment each cycle is the everyday case.

The Tacoma Narrows bridge collapse of 1940 is the standard cautionary example, though the mechanism was aeroelastic flutter rather than simple resonance — a distinction often lost in the retelling.

Common pitfalls

• Thinking amplitude changes the period. It does not, for true SHM.
• Putting the mass into the pendulum period. It cancels.
• Using $T = 2\pi\sqrt{L/g}$ for large swings.
• Confusing $\omega$ (rad/s) with $f$ (Hz). They differ by $2\pi$.
• Assuming maximum speed and maximum acceleration happen at the same place. They are at opposite points.

Practice

Work each fully before opening the answer. Original exercises written for this guide in the style of the course's problem sets.

Practice 1 · Period

A $2$ kg mass on a $50$ N/m spring. Find the period.

Answer

$T = 2\pi\sqrt{2/50} = 1.26$ s.

Practice 2 · Pendulum length

How long is a pendulum with a $2$ s period?

Answer

$L = gT^2/(4\pi^2) = 0.99$ m — the classic one-metre seconds pendulum.

Practice 3 · Where is it fastest

At what point of the motion is speed greatest?

Answer

At equilibrium, $x = 0$, where the force and acceleration are zero.

Practice 4 · Doubling amplitude

What happens to period and energy if amplitude doubles?

Answer

Period unchanged; energy quadruples, since $E \propto A^2$.

Practice 5 · Maximum speed

A $0.3$ kg mass on a $120$ N/m spring oscillates with $A = 0.05$ m. Find $v_{\max}$.

Solution

$\omega = \sqrt{120/0.3} = 20$ rad/s, so $v_{\max} = A\omega = 1$ m/s.

Practice 6 · Pendulum on the Moon

A pendulum with a $1$ s period on Earth is taken to the Moon, where $g$ is $1.6$ m/s$^2$. Find its new period.

Solution

$T \propto 1/\sqrt{g}$, so the period scales by $\sqrt{9.8/1.6} = 2.47$. The new period is $2.5$ s.

Practice 7 · Energy split

At what displacement is the energy half kinetic and half potential?

Solution

Set $\tfrac12kx^2 = \tfrac12(\tfrac12kA^2)$, giving $x = A/\sqrt2 \approx 0.71A$ — further out than halfway.

Practice 8 · Vertical spring

Does hanging a mass on a vertical spring change its period compared with a horizontal one?

Answer

No. Gravity shifts the equilibrium position down by $mg/k$, but the restoring force about that new equilibrium is still $-kx$, so $T = 2\pi\sqrt{m/k}$ is unchanged.

Expanded topic checklist

Course coverage. This unit is part of the complete PHY131H1 topic map. It should cover the following items before you move on:

• The SHM condition. $F = -kx$ and what the minus sign encodes.
• Period and frequency. Spring and pendulum formulas, and what each does not depend on.
• Isochronism. Why amplitude cancels, and where the pendulum approximation breaks.
• Energy. The exchange between $K$ and $U$, damping regimes, and resonance.

Textbook/source alignment

The calendar lists "oscillations" here. Check the syllabus on Quercus for assigned problems; this material leads directly into waves in Unit 10.

Core idea map

A linear restoring force gives sinusoidal motion with a period set only by $m$ and $k$. Energy sloshes between kinetic and potential. Unit 10 lets that oscillation travel.

Study routine. Identify $k$ and $m$ — or $L$ and $g$ — first, and the period follows immediately. Most oscillation questions are one substitution once those are named.

Deep practice extensions

Use these after the main practice set. They push on definitions and edge cases rather than routine calculation.

Extension 1 · The condition

What force law produces SHM?

Guide

$F = -kx$: restoring, and proportional to displacement. Linearity is what makes the period amplitude-independent.

Extension 2 · What the period depends on

Does the pendulum period depend on mass?

Guide

No. It depends only on length and $g$. The mass cancels, since both the restoring force and the inertia scale with it.

Extension 3 · Extremes

Where are speed and acceleration each greatest?

Guide

Speed at equilibrium, acceleration at maximum displacement — opposite points.

Extension 4 · Damping choice

Why is a car suspension critically damped?

Guide

To return to level as fast as possible without oscillating. Underdamped would bounce; overdamped would feel sluggish over bumps.

← Unit 08Rotation & angular momentum Unit 10 →Waves & sound