Simple Harmonic Motion

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Simple Harmonic Motion — Key Facts SHM definition: $a = -\omega^2 x$ where $a$ is acceleration, $x$ is displacement, $\omega$ is angular frequency Position: $x = A\cos(\omega t + \phi)$; velocity: $v = -A\omega\sin(\omega t + \phi)$; acceleration: $a = -A\omega^2\cos(\omega t + \phi)$ Angular frequency: $\omega = 2\pi f = \frac{2\pi}{T}$ Period: $T = 2\pi\sqrt{\frac{m}{k}}$ for spring-mass; $T = 2\pi\sqrt{\frac{l}{g}}$ for simple pendulum ⚡ Exam tip: In SHM, acceleration is always opposite to displacement (restoring force)

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Simple Harmonic Motion — JAMB Physics Study Guide

Spring-mass system: Restoring force: $F = -kx$ where $k$ is spring constant Period: $T = 2\pi\sqrt{\frac{m}{k}}$; frequency: $f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$ Note: Period is independent of amplitude (isochronous property)

Simple pendulum: Restoring force: $F = -\frac{mg}{l}x$ for small angles (where $x$ is arc length) Period: $T = 2\pi\sqrt{\frac{l}{g}}$; valid for $\theta < 10°$ Effective length = distance from pivot to centre of mass

Energy in SHM: Potential energy: $PE = \frac{1}{2}kx^2 = \frac{1}{2}kA^2\cos^2(\omega t + \phi)$ Kinetic energy: $KE = \frac{1}{2}mv^2 = \frac{1}{2}kA^2\sin^2(\omega t + \phi)$ Total energy: $E = \frac{1}{2}kA^2$ = constant (no damping)

Damped oscillations: In real systems, damping occurs; amplitude decreases: $A = A_0 e^{-bt/2m}$ Critical damping: returns to equilibrium fastest without oscillating Overdamping: returns slowly without oscillating Underdamping: oscillates with decreasing amplitude

Common student mistakes: confusing $\omega$ with angular velocity; using $T = 2\pi\sqrt{l/g}$ for non-small angles; forgetting that period of pendulum is independent of mass.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Simple Harmonic Motion — Comprehensive Physics Notes

Derivation of SHM equation: For a mass-spring system: $F = -kx = ma$ $$m\frac{d^2x}{dt^2} = -kx$$ $$\frac{d^2x}{dt^2} + \frac{k}{m}x = 0$$

Solution: $x = A\cos(\omega t + \phi)$ where $\omega^2 = k/m$

Phase relationships:

Quantity	Phase relative to x
Velocity $v$	Leads by $\pi/2$ (90°)
Acceleration $a$	Leads by $\pi$ (180°) / in anti-phase

At mean position: $x = 0$, $v = v_{max} = A\omega$, $a = 0$, $KE = \frac{1}{2}mA^2\omega^2$, $PE = 0$ At extreme: $x = \pm A$, $v = 0$, $a = \pm a_{max} = \pm A\omega^2$, $KE = 0$, $PE = \frac{1}{2}kA^2$

Compound pendulum: $T = 2\pi\sqrt{\frac{I}{mgd}}$ where $I$ is moment of inertia about pivot, $d$ is distance from pivot to centre of mass. Radius of gyration: $k = \sqrt{I/m}$; then $T = 2\pi\sqrt{\frac{k^2}{gd}} = 2\pi\sqrt{\frac{l_{eq}}{g}}$ Where $l_{eq} = I/(md)$ is the equivalent simple pendulum length.

Oscillations in fluids: For a body floating in liquid (oscillating vertically): $T = 2\pi\sqrt{\frac{V}{gA}}$ where $V$ is submerged volume, $A$ is cross-sectional area.

SHM and uniform circular motion: SHM is the projection of uniform circular motion onto a diameter. If a point moves in a circle of radius $A$ with angular speed $\omega$, its projection on any diameter executes SHM with amplitude $A$ and angular frequency $\omega$.

JAMB exam patterns:

• 2023 JAMB: A simple pendulum has period 2 s; find new period if length is increased 4 times
• 2022 JAMB: At what position in SHM is kinetic energy maximum and potential energy minimum?
• 2021 JAMB: A mass-spring system has period 0.5 s; find spring constant if mass is 0.1 kg
• 2020 JAMB: Effect of doubling amplitude on period of simple pendulum

Important comparisons:

System	Period formula	影响因素
Spring-mass	$T = 2\pi\sqrt{m/k}$	mass, spring constant
Simple pendulum	$T = 2\pi\sqrt{l/g}$	length, gravity (NOT mass)
Compound pendulum	$T = 2\pi\sqrt{I/(mgd)}$	moment of inertia, centre of mass
Liquid in U-tube	$T = 2\pi\sqrt{h/(2g)}$	height of liquid

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📊 JAMB Exam Essentials

Detail	Value
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Subjects	4 subjects (language + 3 for course)
Time	2 hours
Marking	+1 per correct answer
Score	400 max (used for university admission)
Registration	January – February each year

🎯 High-Yield Topics for JAMB

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📝 Previous Year Question Patterns

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💡 Pro Tips

• Use of English carries the most weight — master grammar rules and comprehension strategies
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• Past questions are highly predictive — repeat patterns appear every year
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🔗 Official Resources

• JAMB Official
• JAMB Syllabus

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