Class 11 · Chapter 14

Oscillations

Overview, notes, short notes, formula sheet, daily practice problems, previous year questions, and videos for this chapter — all in one place.

Oscillations Formula Sheet

9 formulas across 3 topics in Oscillations.

1 min read

Updated 2026-07-04 · v1.0.0

SHM — Equation of Motion & Kinematics

SHM is defined by one property: restoring acceleration proportional to displacement, a = −ω²x. Spot that, and T = 2π/ω follows instantly.

a = −ω²x; x = A sin(ωt + φ); T = 2π/ω

Defining equation and solution of SHM

EasyAsked very oftenJEE MainJEE AdvNEETMHT-CETBoards

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
ω	angular frequency	rad/s	[T⁻¹]
A	amplitude	m	[L]
φ	initial phase	rad	[M⁰L⁰T⁰]
T	time period	s	[T]

Valid when

a ∝ −x is the TEST for SHM — check it before quoting any period formula

Common mistakes

Time from mean to extreme is T/4, but mean to A/2 takes only T/12 — SHM is fast near the mean, slow near the extremes

Read the concept →

v = ω√(A² − x²); a = −ω²x

Velocity and acceleration at displacement x

EasyAsked very oftenJEE MainNEETMHT-CETBoards

Differentiating x(t), then eliminating t

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
v	speed at displacement x	m/s	[LT⁻¹]
x	displacement from the mean position	m	[L]

Valid when

v(max) = ωA at the mean position; a(max) = ω²A at the extremes
v and a are 90° out of phase; a and x are 180° out of phase

Common mistakes

At the extreme, v = 0 but a is MAXIMUM — zero speed never means zero acceleration in SHM

Read the concept →

Worth remembering

SHM is the projection of uniform circular motion on a diameter — the reference-circle picture solves every phase/timing question

Oscillating Systems — Springs & Pendulums

Every system reduces to finding its ω. Spring: √(k/m). Pendulum: √(g/L). Everything else is variations on these two.

T = 2π √(m/k)

Period of a spring-mass system

EasyAsked very oftenJEE MainJEE AdvNEETMHT-CETBoards

F = −kx compared with F = −mω²x

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
m	oscillating mass	kg	[M]
k	spring constant	N/m	[MT⁻²]

Valid when

Independent of gravity — the same on Earth, in a lift, or in orbit
Vertical spring: gravity only SHIFTS the mean position, period unchanged

Common mistakes

Cutting a spring into n equal parts makes each piece n times STIFFER (k' = nk), so the period drops by √n

Read the concept →

Series: 1/k(eq) = 1/k₁ + 1/k₂; Parallel: k(eq) = k₁ + k₂

Combination of springs

MediumAsked very oftenJEE MainNEETMHT-CET

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
k(eq)	equivalent spring constant	N/m	[MT⁻²]

Valid when

OPPOSITE of resistors: springs side-by-side (or one on each side of the mass) ADD; end-to-end combine reciprocally

Common mistakes

A mass between two springs attached to opposite walls is a PARALLEL combination — both springs push it back together

Read the concept →

T = 2π √(L/g)

Period of a simple pendulum

EasyAsked very oftenJEE MainJEE AdvNEETMHT-CETBoards

Restoring torque −mgL sinθ ≈ −mgLθ for small angles

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
L	length from pivot to the bob's centre	m	[L]
g	effective gravitational acceleration	m/s²	[LT⁻²]

Valid when

Small angles only (sinθ ≈ θ)
Independent of the bob's mass
In an accelerating frame use g(eff): lift up → g + a; free fall → T infinite (no oscillation)

Common mistakes

The seconds pendulum has T = 2 s (not 1 s) — one second per HALF swing, L ≈ 1 m on Earth

Read the concept →

T = 2π √(I/(mgd))

Period of a physical (compound) pendulum

MediumAsked sometimesJEE MainJEE AdvNEET

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
I	moment of inertia about the pivot	kg·m²	[ML²]
d	distance from pivot to the centre of mass	m	[L]

Valid when

Reduces to the simple pendulum when I = mL² and d = L
Rod pivoted at one end: T = 2π√(2L/3g)

Read the concept →

Worth remembering

A floating cylinder pushed down, liquid in a U-tube, and a ball in a spherical bowl all execute SHM — derive ω from a = −ω²x each time rather than memorising

NEET repeats pendulum-in-lift and spring-cutting; JEE Advanced favours energy-method derivations of ω for unusual systems.

Energy in SHM, Damping & Resonance

Total energy in SHM is constant and proportional to A² — kinetic and potential merely trade places twice every period.

E = ½kA² = ½mω²A²; KE = ½mω²(A² − x²); PE = ½mω²x²

Energy in SHM

EasyAsked very oftenJEE MainJEE AdvNEETMHT-CETBoards

KE + PE at any x, using v = ω√(A² − x²)

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
E	total mechanical energy	J	[ML²T⁻²]
A	amplitude	m	[L]

Valid when

KE = PE at x = A/√2, not at A/2
KE and PE each oscillate at TWICE the SHM frequency

Common mistakes

Doubling the amplitude quadruples the energy — E ∝ A²

Read the concept →

x = A e^(−bt/2m) cos(ω't + φ)

Damped oscillation

MediumAsked sometimesJEE MainNEET

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
b	damping constant (F(damp) = −bv)	kg/s	[MT⁻¹]
ω'	damped angular frequency √(k/m − b²/4m²)	rad/s	[T⁻¹]

Valid when

Amplitude decays exponentially; energy decays as e^(−bt/m) — twice as fast as amplitude

Read the concept →

Resonance: driving frequency = natural frequency (ω(d) = ω₀)

Forced oscillations and resonance

EasyAsked sometimesJEE MainNEETMHT-CETBoards

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
ω(d)	driving (forcing) frequency	rad/s	[T⁻¹]
ω₀	natural frequency of the system	rad/s	[T⁻¹]

Valid when

Steady state: the system oscillates at the DRIVING frequency, not its own
Amplitude peaks at resonance; lighter damping → sharper, taller peak

Read the concept →

Worth remembering

Soldiers break step on bridges to avoid resonance — the Tacoma Narrows collapse is the textbook cautionary tale

Stuck on a concept in Oscillations?

Message Ajay Sir directly on WhatsApp for doubt support on this chapter.

Ask a doubt

Previous chapter13. Kinetic Theory of Gases Next chapter 15. Waves

Back to Class 11 chapters