PiTheory

PiTheory JEE Physics Practice & Revision Pack · Class 11 · Chapter 13 · JEE Main

Oscillations

Chapter-wise practice and revision for JEE Main: concise notes on SHM equations and phase, energy in SHM, spring systems, pendulums and floating bodies, and damped oscillations, concept figures, and 50 selected previous-year JEE Main questions covering the major concepts and question patterns, each with a step-by-step solution.

50 selected questions5 topics6 concept figuresPreview · first 3 of 50 questions

How to study this chapter

  1. Look at the topic's figures, then read its notes (5 to 10 minutes). They give the major equations and methods the questions use.
  2. Solve the topic's questions with a timer: 2 minutes for Easy, 3 to 4 for Medium and Hard.
  3. Check the solution. Read the Point to note line even if you were right: it is the idea you will reuse.
  4. Mark each question Got it or Revise. Come back to the Revise list after two days.

Where the marks come from

Main idea of each of the 122 previous-year JEE questions in the PiTheory bank

SHM basics42
Springs32
Energy24
Pendulums21
Damped3

Contents

  1. 1SHM: displacement, velocity, acceleration and phaseQ1–Q14
  2. 2Energy in SHMQ15–Q24
  3. 3Spring–mass systemsQ25–Q38
  4. 4Pendulums, physical pendulums and floating bodiesQ39–Q47
  5. 5Damped oscillationsQ48–Q50
  6. ✓Last-day checklist
01

Topic 1 · Q1–Q14

SHM: displacement, velocity, acceleration and phase

42 of the 122 JEE PYQs in the bank test this as their main idea

Look first

θPxAAω−A+Ax = A sin(ωt + φ)v = Aω cos(ωt + φ)a = −ω²xθ = ωt + φ = phase
SHM is the shadow of uniform circular motion.
xvaT/4T/2Tv leads x by π/2a opposite to x
x, v and a are a quarter period apart.

Quick notes

$$F=-kx\ \Rightarrow\ a=-\omega^2x,\qquad x=A\sin(\omega t+\phi),\qquad \omega=\frac{2\pi}{T}=2\pi f$$

SHM is the projection of uniform circular motion (radius $A$, angular speed $\omega$) on a diameter. The phase $\omega t+\phi$ is the angle on that reference circle.

QuantityFormulaAt meanAt extreme
Velocity$v=\omega\sqrt{A^2-x^2}$$A\omega$ (max)$0$
Acceleration$a=-\omega^2x$$0$$\omega^2A$ (max)
$v$–$x$ graph$\dfrac{x^2}{A^2}+\dfrac{v^2}{A^2\omega^2}=1$ellipse
FromToPhase turnedTime
$0$$A/2$$\pi/6$$T/12$
$0$$A/\sqrt2$$\pi/4$$T/8$
$0$$\sqrt3A/2$$\pi/3$$T/6$
$0$$A$$\pi/2$$T/4$
  • Two readings: $|a|=\omega^2x$ gives $\omega$; $A^2=x^2+\dfrac{v^2}{\omega^2}$ gives the amplitude. From $(x_1,v_1)$, $(x_2,v_2)$: $\omega^2=\dfrac{v_1^2-v_2^2}{x_2^2-x_1^2}$.
  • Initial phase: from $x(0)$ and the sign of $v(0)$.
  • Sudden change of speed at a point: same $x$, new $A'^2=x^2+v'^2/\omega^2$.
  • Superposition: $a\sin\theta+b\cos\theta=\sqrt{a^2+b^2}\sin(\theta+\phi)$; same-frequency SHMs add like vectors, $R^2=A_1^2+A_2^2+2A_1A_2\cos\delta$.
  • Is it SHM? One sinusoid (plus a constant) only. $\sin^2\omega t$ is SHM with $2\omega$; $\sin^3$, $\cos\omega t+\cos2\omega t$ are periodic but not SHM; incommensurate frequencies are not periodic.
  • Distance in time $t$: count whole periods ($4A$ each), then use phase for the rest.
Traps. Time is not proportional to distance (the particle is slow near the ends). Phase in radians. "$\tfrac58$ oscillation" means $\tfrac58$ of the path $4A$.

Practice: first 3 of 14 questions

Q1JEE Main 2023 (10 Apr, Shift 1)Easy

A particle executes S.H.M. of amplitude $A$ along $x$-axis. At $t=0$, the position of the particle is $x=\dfrac A2$ and it moves along positive $x$-axis. The displacement of particle in time $t$ is $x=A\sin(\omega t+\delta)$, then the value of $\delta$ will be

  1. A$\dfrac\pi2$
  2. B$\dfrac\pi6$
  3. C$\dfrac\pi3$
  4. D$\dfrac\pi4$
Answer and solution

Answer: Option B: $\dfrac\pi6$

Concept. The initial phase is fixed by the starting position and the direction of motion. Put $t=0$ in $x=A\sin(\omega t+\delta)$ and in $v=A\omega\cos(\omega t+\delta)$.

Step 1. Position. $\dfrac A2=A\sin\delta\Rightarrow\sin\delta=\dfrac12$, so $\delta=\dfrac\pi6$ or $\dfrac{5\pi}6$.

Step 2. Direction. Moving along $+x$ needs $v>0$, i.e. $\cos\delta>0$, which picks $\delta=\dfrac\pi6$.

Answer. Option B. ($\dfrac{5\pi}{6}$ would be the same point while moving towards the mean.)

Point to note. Position and direction fix $\delta$.

Q2JEE Main 2026 (2 Apr, Shift 2)Easy

The equation of motion of a particle is given by $x=a\sin(50t+\pi/3)$ cm. The particle will come to rest at time $t_1$ and it will have zero acceleration at time $t_2$. The $t_1$ and $t_2$ respectively are ______.

  1. A$\dfrac{\pi}{300}$ s, $\dfrac{\pi}{75}$ s
  2. B$\dfrac{\pi}{75}$ s, $\dfrac{\pi}{300}$ s
  3. C$\dfrac{\pi}{300}$ s, $\dfrac{\pi}{25}$ s
  4. D$\dfrac{\pi}{50}$ s, $\dfrac{\pi}{100}$ s
Answer and solution

Answer: Option A: $\dfrac{\pi}{300}$ s, $\dfrac{\pi}{75}$ s

Concept. In $x=a\sin\phi$ with phase $\phi=50t+\dfrac\pi3$, the particle is at rest at an extreme ($\phi=\dfrac\pi2$) and has zero acceleration at the mean ($\phi=\pi$, the next mean position after $t=0$).

Step 1. First rest. $50t_1+\dfrac\pi3=\dfrac\pi2\Rightarrow t_1=\dfrac{\pi}{300}$ s.

Step 2. First zero acceleration. $50t_2+\dfrac\pi3=\pi\Rightarrow t_2=\dfrac{2\pi}{150}=\dfrac{\pi}{75}$ s.

Answer. Option A.

Point to note. Rest at phase $\pi/2$, $a=0$ at $\pi$.

Q3JEE Main 2022 (27 Jun, Shift 1)Easy

The displacement of simple harmonic oscillator after $3$ seconds starting from its mean position is equal to half of its amplitude. The time period of harmonic motion is

  1. A$12$ s
  2. B$8$ s
  3. C$36$ s
  4. D$6$ s
Answer and solution

Answer: Option C: $36$ s

Concept. Starting from the mean, $x=A\sin\omega t$. Find the phase at which $x=\dfrac A2$ and convert phase to time with $\omega=\dfrac{2\pi}{T}$.

Step 1. Phase. $\sin\omega t=\dfrac12\Rightarrow\omega t=\dfrac\pi6$.

Step 2. Time. $\dfrac{2\pi}{T}\times3=\dfrac\pi6\Rightarrow T=36$ s.

Answer. Option C. Useful fact: mean to $A/2$ takes $T/12$.

Point to note. Mean to $A/2$: $T/12$.

47 more questions and 4 more topics in the full chapter

  1. 2Energy in SHM10 questions with solutions
  2. 3Spring–mass systems14 questions with solutions
  3. 4Pendulums, physical pendulums and floating bodies9 questions with solutions
  4. 5Damped oscillations3 questions with solutions
  5. ✓Last-day checklist

The complete Oscillations chapter is in the Class 11 Pack and the Complete Pack, as a PDF on WhatsApp. Want to see a whole chapter first? Laws of Motion and Current Electricity are free in full.

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