Class 11 Physics · NCERT notes for NEET

Oscillations: NCERT Notes, Formulas and Tricks for NEET

Periodic motion, simple harmonic motion (SHM), the energy of an oscillator, and the spring and pendulum systems. NEET usually asks a direct question from here (a time period, a speed at some x, or an energy ratio), and the same ideas carry straight into Waves.

1. Simple harmonic motion
2. Energy in SHM 🔒
3. Spring and pendulum systems 🔒

Topic 1 is open for everyone. The full chapter, quick revision and practice open with a plan, or start free with 5 full chapters.

Open the interactive version (simulations and tap-to-answer quick checks).

Topic 1

Simple harmonic motion

A swing, a guitar string, the needle of a sewing machine and the atoms in a crystal all obey one equation: $a = - ω^{2} x$ .

Periodic motion repeats after a fixed time. Oscillatory motion is a to-and-fro motion about a mean (equilibrium) position.
Every oscillatory motion is periodic, but not every periodic motion is oscillatory. Uniform circular motion and planetary orbits are periodic but not oscillatory.
Period $T$ is the smallest time after which the motion repeats. Frequency $ν = 1/ T$ (unit hertz, $1 Hz = 1 s^{- 1}$ ). Angular frequency $ω = 2 π ν = 2 π / T$ in rad/s.
SHM: the restoring force is proportional to the displacement and always points towards the mean position, $F = - k x$ . Then $a = - ω^{2} x$ with $ω = k / m$ .
Displacement $x (t) = A cos (ω t + ϕ)$ . $A$ is the amplitude, $(ω t + ϕ)$ is the phase and $ϕ$ is the phase constant. $A$ and $ϕ$ are fixed by the initial position and velocity.
$A sin ω t + B cos ω t = D sin (ω t + ϕ)$ with $D = A^{2} + B^{2}$ and $tan ϕ = B / A$ . It is still SHM with the same period. Any periodic function can be built from sines and cosines of different periods.
SHM is the projection of uniform circular motion on any diameter of the reference circle. Radius = $A$ , angular speed = $ω$ .
Velocity $v = - ω A sin (ω t + ϕ)$ , so $∣ v ∣ = ω A^{2} - x^{2}$ : maximum $ω A$ at the mean, zero at the extremes.
Acceleration $a = - ω^{2} A cos (ω t + ϕ) = - ω^{2} x$ : maximum $ω^{2} A$ at the extremes, zero at the mean. Velocity leads $x$ by $π /2$ ; acceleration leads $x$ by $π$ .
The period does not depend on the amplitude (isochronous). In one period the particle travels $4 A$ .

P moves on the reference circle at constant omega; its foot P′ on the diameter does SHM with amplitude equal to the radius. — P moves on the reference circle at constant $ω$ ; its foot P′ on the diameter does SHM with amplitude equal to the radius.

Read the three graphs at the same instant: when x = 0, speed is maximum and a = 0; when x = pm A, v = 0 and |a| is maximum. — Read the three graphs at the same instant: when $x = 0$ , speed is maximum and $a = 0$ ; when $x = \pm A$ , $v = 0$ and $∣ a ∣$ is maximum.

Acceleration always points to the mean. The a-x graph is a straight line through the origin with negative slope -omega^2: that is the test for SHM. — Acceleration always points to the mean. The $a$ - $x$ graph is a straight line through the origin with negative slope $- ω^{2}$ : that is the test for SHM.

Simulation: SHM and the reference circleTry it

Formulas

$x = A cos (ω t + ϕ)$
Displacement in SHM
$ω = \frac{2 π}{T} = 2 π ν = \frac{k}{m}$
Angular frequency in rad/s
$v = \pm ω A^{2} - x^{2}, v_{m a x} = ω A$
Speed at displacement $x$
$a = - ω^{2} x, a_{m a x} = ω^{2} A$
Defining condition of SHM
$F = - k x, k = m ω^{2}$
Restoring force; $k$ in N/m
$A = x_{0}^{2} + (\frac{v _{0}}{ω})^{2}$
Amplitude from initial position $x_{0}$ and velocity $v_{0}$
$\frac{v ^{2}}{ω ^{2} A ^{2}} + \frac{x ^{2}}{A ^{2}} = 1$
The $v$ - $x$ graph of SHM is an ellipse

Tips, tricks and hacks

Ratio gives omega

a_{m a x} / v_{m a x} = ω

and

v_{m a x}^{2} / a_{m a x} = A

. Example:

v_{m a x} = 2

m/s,

a_{m a x} = 8

m/s

^{2}

gives

ω = 4

rad/s and

A = 0.5

m in one line.

Spot SHM fastWrite

a

(or

F

) in terms of

x

. If it is

- (positive constant) \times x

, it is SHM and

ω = constant

. Example:

a = - 16 x

gives

ω = 4

rad/s,

T = π /2

Time from the circleUse the reference circle: going from the mean to

x

takes

t = \frac{1}{ω} sin^{- 1} (x / A)

. Mean to

A /2

3 0^{\circ}

of turn, so

t = T /12

;

A /2

A

takes

T /6

Two speeds, two positionsGiven

v_{1}

x_{1}

and

v_{2}

x_{2}

ω^{2} = \frac{v _{1}^{2} - v _{2}^{2}}{x _{2}^{2} - x _{1}^{2}}

. Then

A^{2} = x_{1}^{2} + v_{1}^{2} / ω^{2}

. No need to solve two equations each time.

Common mistakes

✗ Velocity and acceleration are both maximum at the same point.

✓ Speed is maximum at the mean ( $x = 0$ ); acceleration is maximum at the extremes. They are $π /2$ out of phase.

✗ A bigger swing (larger $A$ ) takes more time.

✓ In SHM $T = 2 π / ω$ does not contain $A$ . Larger $A$ just means larger $v_{m a x}$ .

✗ Every periodic motion, like uniform circular motion, is SHM.

✓ SHM needs to-and-fro motion with $a = - ω^{2} x$ . Circular motion is periodic, not oscillatory; only its projection is SHM.

✗ In time $T /8$ from the mean, the particle covers $A /2$ .

✓ Motion is not uniform. In $T /8$ (phase $4 5^{\circ}$ ) it covers $A sin 4 5^{\circ} = A / 2$ starting from the mean.

NEET traps

$sin^{2} ω t$ repeats with period $π / ω$ , not $2 π / ω$ . NEET 2018's answer: it is periodic, but not SHM (its acceleration is not $\propto - x$ about $x = 0$ ).

$sin ω t + cos 2 ω t$ is periodic but NOT SHM; $sin ω t + cos ω t$ IS SHM with amplitude $2$ . $e^{- ω t}$ is not periodic at all.

Phase difference between displacement and acceleration is $π$ (not $π /2$ ); between velocity and acceleration it is $π /2$ .

Quick check: A particle in SHM has amplitude 10 cm and maximum speed 2 m/s. Its speed when it is 6 cm from the mean position is

1.2 m/s
1.6 m/s
0.8 m/s
1.8 m/s

Answer: B. $ω = 2/0.1 = 20$ rad/s, so $v = 20 0.1 0^{2} - 0.0 6^{2} = 20 \times 0.08 = 1.6$ m/s.

Topic 2 · 🔒

Energy in SHM

Key points, graphs, every formula, tricks, common mistakes and NEET traps for this topic are inside Dhruv.

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Topic 3 · 🔒

Spring and pendulum systems

Key points, graphs, every formula, tricks, common mistakes and NEET traps for this topic are inside Dhruv.

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Practise Oscillations questions

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