Oscillations (Simple Harmonic Motion) — Complete Formula

Basic Definition of SHM

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
SHM Condition	[a = -\omega^2 x]	[a] = acceleration, [\omega] = angular frequency, [x] = displacement	m/s²	Defining equation of SHM ⭐
Angular Frequency	[\omega = 2\pi f]	[f] = frequency	rad/s	Basic relation ⭐
Time Period	[T = \dfrac{2\pi}{\omega}]	[T] = time period	s	Inverse of frequency
Frequency	[f = \dfrac{1}{T}]	—	Hz	Cycles per second

💡 Memory Hint:
SHM → restoring force proportional to displacement

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Displacement	[x = A\sin(\omega t + \phi)]	[A] = amplitude, [\phi] = phase constant	m	General SHM equation ⭐
Velocity	[v = \omega\sqrt{A^2 – x^2}]	[v] = velocity	m/s	Max at mean position ⭐
Acceleration	[a = -\omega^2 x]	—	m/s²	Opposite direction ⚠️
Maximum Velocity	[v_{max} = A\omega]	—	m/s	At mean position ⭐
Maximum Acceleration	[a_{max} = A\omega^2]	—	m/s²	At extreme position ⭐

💡 Memory Hint:

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Total Energy	[E = \dfrac{1}{2}kA^2]	[k] = force constant	J	Constant ⭐
Kinetic Energy	[K = \dfrac{1}{2}k(A^2 – x^2)]	—	J	Max at mean position
Potential Energy	[U = \dfrac{1}{2}kx^2]	—	J	Max at extreme ⭐

💡 Memory Hint:
Energy shifts between KE ↔ PE

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Time Period	[T = 2\pi\sqrt{\dfrac{m}{k}}]	[m] = mass, [k] = spring constant	s	Most important ⭐

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Time Period	[T = 2\pi\sqrt{\dfrac{l}{g}}]	[l] = length, [g] = gravity	s	Independent of mass ⭐
Frequency	[f = \dfrac{1}{2\pi}\sqrt{\dfrac{g}{l}}]	—	Hz	Derived form

💡 Memory Hint:
Pendulum → depends only on length & gravity

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Phase	[\theta = \omega t + \phi]	[\theta] = phase	rad	Position in cycle
Phase Difference	[\Delta \phi = \omega \Delta t]	—	rad	Time shift relation ⭐

💡 Memory Hint:
Phase tells where particle is in motion

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
SHM Projection	Projection of circular motion	—	—	Key concept ⭐
Velocity Relation	[v = \omega\sqrt{A^2 – x^2}]	—	m/s	Derived from circle

💡 Memory Hint:
SHM = projection of uniform circular motion

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Force	[F = -kx]	[F] = restoring force	N	Hooke’s law ⭐
Angular Frequency	[\omega = \sqrt{\dfrac{k}{m}}]	—	rad/s	Derived from motion ⭐
Acceleration Relation	[a = -\omega^2 x]	—	m/s²	Core identity

💡 Memory Hint:
Everything revolves around [\omega]

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