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Oscillations NCERT Solutions, Complete PDF Content and Important Questions
NCERT Solutions for Class 11 Physics Chapter 13 Oscillations PDF Download (2026) + Important Questions Welcome to the rhythmic world of physics! Have you ever stared at a swinging pendulum clock or watched the suspension of a car bounce after hitting a bump? That mesmerizing back-and-forth motion is exactly what we are going to study in CBSE Class 11 Physics Chapter 13: Oscillations . For students preparing for their 2026 board exams or aiming to crack competitive exams like JEE and NEET, mastering Oscillations is absolutely essential. This chapter forms the critical bridge between mechanics and wave physics. If you understand how a simple spring oscillates, you will easily grasp complex concepts like sound waves, alternating current (AC), and even quantum mechanics later on! In this comprehensive guide, we will break down the entire chapter into simple, student-friendly concepts.
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ChapterOscillations
SubjectPhysics
Class11
BoardCBSE
DifficultyModerate to High
Exam Weightage~5-7 Marks
Learning Objectives
- Differentiate between periodic motion, oscillatory motion, and Simple Harmonic Motion (SHM).
- Understand the kinematics of SHM (displacement, velocity, and acceleration) using circular motion projections.
- Calculate the kinetic, potential, and total mechanical energy of an oscillating system.
- Derive the time period of a simple pendulum and a spring-mass system.
- Understand the real-world effects of damping and resonance (forced oscillations).
Key Concepts, Definitions and Formulas
- Periodic Motion: Motion that repeats itself at regular intervals of time (e.g., planets revolving around the
- Sun). Oscillatory Motion: A special type of periodic motion where a body moves back and forth about a fixed mean position (e.g., a swinging pendulum)
- Simple Harmonic Motion (SHM): The simplest form of oscillatory motion where the restoring force is directly proportional to the displacement from the mean position and acts oppositely to it
- Force Equation: $F = -k x$ (where $k$ is the force constant)
- Kinematics of SHM: Displacement ($x$): $x(t) = A \cos(\omega t + \phi)$ (where $A$ is amplitude, $\omega$ is angular frequency, $\phi$ is phase angle). Velocity ($v$): $v(t) = \frac{dx}{dt} = -A\omega \sin(\omega t + \phi) = \pm \omega\sqrt{A^2 - x^2}$. Acceleration ($a$): $a(t) = \frac{dv}{dt} = -\omega^2 x$
- Energy in SHM: Potential Energy ($U$): $U = \frac{1}{2} k x^2 = \frac{1}{2} m \omega^2 x^2$ Kinetic Energy ($K$): $K = \frac{1}{2} m v^2 = \frac{1}{2} m \omega^2 (A^2 - x^2)$ Total Energy ($E$): $E = U + K = \frac{1}{2} k A^2$ (Total energy remains constant!). Time Period ($T$): The time taken for one complete oscillation. $T = \frac{2\pi}{\omega}$
- Spring-Mass System: $T = 2\pi \sqrt{\frac{m}{k}}$
- Simple Pendulum: $T = 2\pi \sqrt{\frac{l}{g}}$
- Damped Oscillations: Oscillations where the amplitude decreases over time due to resistive forces (like friction or air drag)
- Resonance: A phenomenon that occurs when the frequency of an external driving force matches the natural frequency of the system, resulting in a dramatic increase in amplitude. To really grasp how energy shifts during SHM, let's explore this interactive Simple Harmonic Motion simulator!
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Full NCERT Solutions
NCERT Q13.1: Which of the following examples represent periodic motion? (a) A swimmer completing one (return) trip from one bank of a river to the other and back. (b) A freely suspended bar magnet displaced from its N-S direction and released. (c) A hydrogen molecule rotating about its center of mass. (d) An arrow released from a bow.
Answer: (a) Not periodic: While the swimmer returns to the start, the motion lacks a definite, constant time period. (b) Periodic: A freely suspended bar magnet, when displaced, acts like a magnetic pendulum. The Earth's magnetic field provides a restoring torque, making it oscillate periodically. (c) Periodic: A molecule rotating at a constant angular velocity completes rotations in equal intervals of time, making it periodic. (d) Not periodic: An arrow released from a bow travels forward and does not return to its starting position.
NCERT Q13.2: Which of the following examples represent (nearly) simple harmonic motion and which represent periodic but not simple harmonic motion? (a) the rotation of earth about its axis. (b) motion of an oscillating mercury column in a U-tube. (c) motion of a ball bearing inside a smooth curved bowl, when released from a point slightly above the lower most point. (d) general vibrations of a polyatomic molecule about its equilibrium position.
Answer: (a) Periodic but NOT SHM: The Earth rotates at a constant rate (periodic), but it does not move back and forth about a mean position, so it is not SHM. (b) SHM: If the mercury column in a U-tube is displaced, the restoring force (weight of the displaced liquid) is proportional to the displacement, making it SHM. (c) SHM: For a very small displacement from the lowest point of a smooth bowl, the restoring component of gravity ($mg \sin\theta$) is nearly proportional to the displacement, executing SHM. (d) Periodic but NOT SHM: A polyatomic molecule has multiple natural frequencies resulting in a complex superposition of waves. It is periodic but not a simple harmonic motion.
NCERT Q13.3: Figure depicts four x-t plots for linear motion of a particle. Which of the plots represent periodic motion? What is the period of motion (in case of periodic motion)? (Assume standard textbook graphs) (a) A non-repeating jagged curve. (b) A perfectly repeating triangular wave pattern with a gap of 2 seconds between peaks. (c) A step-function graph. (d) A repeating semi-circular pattern with a gap of 2 seconds.
Answer: (a) Not periodic: The graph does not repeat its pattern at regular intervals of time. (b) Periodic: The triangular pattern repeats exactly. The time difference between two consecutive identical states (e.g., peaks) is 2 seconds. Period $T = 2 \text{ s}$. (c) Not periodic: (Assuming a continuous forward step function) The particle keeps moving forward and never returns to a previous state. (d) Periodic: The pattern repeats perfectly. The time difference between identical states is 2 seconds. Period $T = 2 \text{ s}$.
NCERT Q13.4: Which of the following functions of time represent (a) simple harmonic, (b) periodic but not simple harmonic, and (c) non-periodic motion? Give period for each case of periodic motion ($\omega$ is any positive constant): (a) $\sin\omega t - \cos\omega t$ (b) $\sin^3\omega t$ (c) $3\cos(\pi/4 - 2\omega t)$ (d) $\cos\omega t + \cos3\omega t + \cos5\omega t$ (e) $\exp(-\omega^2 t^2)$ (f) $1 + \omega t + \omega^2 t^2$
Answer: (a) SHM: Let $y = \sin\omega t - \cos\omega t$. Multiply and divide by $\sqrt{2}$: $y = \sqrt{2} \left(\frac{1}{\sqrt{2}}\sin\omega t - \frac{1}{\sqrt{2}}\cos\omega t\right)$ $y = \sqrt{2} (\cos(\pi/4)\sin\omega t - \sin(\pi/4)\cos\omega t)$ $y = \sqrt{2} \sin(\omega t - \pi/4)$. This is a standard SHM equation. Period $T = 2\pi/\omega$. (b) Periodic but NOT SHM: $\sin^3\omega t = \frac{1}{4}(3\sin\omega t - \sin3\omega t)$. This is a superposition of two SHMs with different frequencies ($\omega$ and $3\omega$). It is periodic but not SHM. Period is $2\pi/\omega$ (the fundamental period). (c) SHM: $y = 3\cos(2\omega t - \pi/4)$ is a simple cosine function representing SHM. The angular frequency is $2\omega$. Period $T = 2\pi / (2\omega) = \pi/\omega$. (d) Periodic but NOT SHM: It is a superposition of three independent SHMs. Period is $2\pi/\omega$. (e) Non-periodic: As $t \to \infty$, the exponential function tends to zero without repeating. (f) Non-periodic: This is a polynomial function that increases continuously to infinity as $t$ increases.
NCERT Q13.5: A particle is in linear simple harmonic motion between two points, A and B, 10 cm apart. Take the direction from A to B as the positive direction and give the signs of velocity, acceleration and force on the particle when it is: (a) at the end A, (b) at the end B, (c) at the mid-point of AB going towards A, (d) at 2 cm away from B going towards A, (e) at 3 cm away from A going towards B, and (f) at 4 cm away from B going towards A.
Answer: Let the mid-point be $O$ (mean position, $x=0$). A is at $x = -5 \text{ cm}$, and B is at $x = +5 \text{ cm}$. Recall: Acceleration $a = -\omega^2 x$. Force $F = -kx$. So, $a$ and $F$ always have the opposite sign to displacement $x$. (a) At A ($x = -5$): Velocity $v = 0$. Since $x$ is negative, acceleration $a$ is positive (+). Force $F$ is positive (+). (b) At B ($x = +5$): Velocity $v = 0$. Since $x$ is positive, acceleration $a$ is negative (-). Force $F$ is negative (-). (c) At mid-point ($x = 0$) towards A: Going towards A means moving in the negative direction, so $v$ is negative (-). $a = 0$ and $F = 0$. (d) At 2 cm from B towards A ($x = +3$): Moving towards A implies negative velocity, $v$ is (-). Since $x$ is positive, $a$ is (-) and $F$ is (-). (e) At 3 cm from A towards B ($x = -2$): Moving towards B implies positive velocity, $v$ is (+). Since $x$ is negative, $a$ is (+) and $F$ is (+). (f) At 4 cm from B towards A ($x = +1$): Moving towards A implies negative velocity, $v$ is (-). Since $x$ is positive, $a$ is (-) and $F$ is (-).
NCERT Q13.6: A spring balance has a scale that reads from 0 to 50 kg. The length of the scale is 20 cm. A body suspended from this balance, when displaced and released, oscillates with a period of 0.6 s. What is the weight of the body?
Answer: Given: Maximum load $m = 50 \text{ kg}$. Maximum extension $x = 20 \text{ cm} = 0.2 \text{ m}$. Time period $T = 0.6 \text{ s}$. Step 1: Find the Spring Constant ($k$) The spring extends by 0.2 m when a load of 50 kg is applied. Restoring force $F = mg = kx$ $k = \frac{mg}{x} = \frac{50 \times 9.8}{0.2} = \frac{490}{0.2} = 2450 \text{ N/m}$. Step 2: Find the mass of the oscillating body ($M$) Formula for time period: $T = 2\pi \sqrt{\frac{M}{k}}$ Squaring both sides: $T^2 = \frac{4\pi^2 M}{k}$ $M = \frac{k T^2}{4\pi^2}$ $M = \frac{2450 \times (0.6)^2}{4 \times (3.14)^2} = \frac{2450 \times 0.36}{39.438} = \frac{882}{39.438} \approx 22.36 \text{ kg}$. Step 3: Find the weight of the body Weight $W = Mg = 22.36 \times 9.8 \approx \textbf{219.1 N}$.
Extra Important Questions for 2026 Exams
Important Q1: The phase difference between displacement and velocity of a particle in SHM is: A) $0$ B) $\pi/2$ C) $\pi$ D) $2\pi$
Answer: B) $\pi/2$ Explanation: If displacement $x = A\sin(\omega t)$, velocity $v = A\omega\cos(\omega t) = A\omega\sin(\omega t + \pi/2)$. The phase difference is $90^\circ$ or $\pi/2$.
Important Q2: A simple pendulum is taken to the Moon. Its time period will: A) Increase B) Decrease C) Remain the same D) Become zero
Answer: A) Increase Explanation: Time period $T = 2\pi\sqrt{l/g}$. Gravity on the Moon is less than on Earth ($g_{moon} = g_{earth}/6$). Since $T \propto 1/\sqrt{g}$, a smaller $g$ means a larger $T$.
Important Q3: Total mechanical energy of a particle executing SHM is proportional to: A) Square of the amplitude B) Amplitude C) Square root of the amplitude D) Inverse of the amplitude
Answer: A) Square of the amplitude Explanation: Total Energy $E = \frac{1}{2} m \omega^2 A^2$. Therefore, $E \propto A^2$.
Important Q4: Can a motion be oscillatory but not simple harmonic? Give an example.
Answer: Yes. A bouncing ball dropped from a height is oscillatory (it goes up and down periodically), but it is not SHM because the restoring force (gravity) is constant and not proportional to the displacement ($F \neq -kx$).
Important Q5: What is the frequency of total kinetic energy for a particle in SHM with frequency $\nu$?
Answer: $2\nu$. In one complete oscillation (cycle), the particle passes through the mean position (maximum kinetic energy) twice. Therefore, the kinetic energy reaches its maximum value twice per cycle.
Important Q6: Define Resonance.
Answer: Resonance is a phenomenon in forced oscillations where the amplitude of oscillation becomes extremely large when the frequency of the external periodic driving force exactly matches the natural frequency of the oscillating body.
Important Q7: Why are soldiers commanded to break step while crossing a suspension bridge?
Answer: If soldiers march in step, the periodic force of their footsteps might accidentally match the natural frequency of the bridge. This would cause resonance, leading to dangerously large oscillations that could collapse the bridge.
Important Q8: Show that the total mechanical energy of a particle executing Simple Harmonic Motion remains constant.
Answer: Consider a particle of mass $m$ executing SHM with amplitude $A$ and angular frequency $\omega$. Displacement $x = A\sin(\omega t)$. Velocity $v = A\omega\cos(\omega t)$. Kinetic Energy ($K$): $K = \frac{1}{2}mv^2 = \frac{1}{2}m(A\omega\cos(\omega t))^2 = \frac{1}{2}m\omega^2 A^2 \cos^2(\omega t)$. Potential Energy ($U$): Work done against restoring force $F = -kx = -m\omega^2 x$. $U = \frac{1}{2}kx^2 = \frac{1}{2}m\omega^2 (A\sin(\omega t))^2 = \frac{1}{2}m\omega^2 A^2 \sin^2(\omega t)$. Total Energy ($E$): $E = K + U = \frac{1}{2}m\omega^2 A^2 \cos^2(\omega t) + \frac{1}{2}m\omega^2 A^2 \sin^2(\omega t)$. $E = \frac{1}{2}m\omega^2 A^2 [\cos^2(\omega t) + \sin^2(\omega t)]$. Since $\sin^2\theta + \cos^2\theta = 1$, $E = \frac{1}{2}m\omega^2 A^2$. This value is a constant, independent of time ($t$) or position ($x$), proving energy is conserved.
Important Q9: Derive the expression for the time period of a simple pendulum.
Answer: Consider a simple pendulum of length $l$ and mass $m$. When displaced by a small angle $\theta$, the restoring force is the tangential component of gravity. Restoring Force $F = -mg \sin\theta$. For very small angles, $\sin\theta \approx \theta$ (in radians). Therefore, $F \approx -mg\theta$. By geometry, arc length (displacement $x$) = $l \cdot \theta \implies \theta = x/l$. Substituting this: $F = -mg (x/l) = -\left(\frac{mg}{l}\right)x$. Comparing this with the standard SHM equation $F = -kx$, we get the effective spring constant $k = \frac{mg}{l}$. Time period $T = 2\pi \sqrt{\frac{m}{k}} = 2\pi \sqrt{\frac{m}{mg/l}} = 2\pi \sqrt{\frac{l}{g}}$.
Important Q10: Two springs of force constants $k_1$ and $k_2$ are connected in series. Find the equivalent force constant and the time period of the mass $m$ attached to it.
Answer: In series, the force ($F$) exerted on both springs is the same, but their total extension ($x$) is the sum of individual extensions ($x_1$ and $x_2$). $x = x_1 + x_2$ Since $F = kx \implies x = F/k$, $\frac{F}{k_{eq}} = \frac{F}{k_1} + \frac{F}{k_2}$ $\frac{1}{k_{eq}} = \frac{1}{k_1} + \frac{1}{k_2} \implies k_{eq} = \frac{k_1 k_2}{k_1 + k_2}$ The time period is $T = 2\pi \sqrt{\frac{m}{k_{eq}}} = 2\pi \sqrt{\frac{m (k_1 + k_2)}{k_1 k_2}}$. A shock absorber in a car consists of a heavy spring and a piston moving in a cylinder filled with oil. This mechanism is designed to absorb the jolts from uneven roads.
Important Q11: What type of oscillation does the car's suspension undergo after hitting a bump?
Answer: It undergoes Damped Oscillation . The oil provides a resistive drag force that decreases the amplitude of oscillation quickly.
Important Q12: If the oil leaks out of the shock absorber, what will happen to the motion of the car after hitting a bump?
Answer: Without the resistive force of the oil, the damping decreases significantly. The car will continue to bounce up and down (oscillate) for a much longer time before coming to rest, causing an uncomfortable ride.
Important Q13: The natural frequency of the car's suspension is 2 Hz. What happens if the car drives over a corrugated road (a road with regular ripples) that hits the tires exactly twice every second?
Answer: This will cause Resonance . The frequency of the external driving force (bumps on the road at 2 Hz) matches the natural frequency of the suspension (2 Hz). The amplitude of the car's bouncing will increase dangerously. (Directions: Select A if both are true and Reason explains Assertion. Select B if both are true but Reason does not explain Assertion. Select C if Assertion is true, Reason is false. Select D if Assertion is false, Reason is true.)
Important Q14: Assertion (A): In Simple Harmonic Motion, the velocity is maximum when the acceleration is zero. Reason (R): Velocity and acceleration have a phase difference of $\pi/2$.
Answer: A . Both are true. At the mean position, displacement is zero, so acceleration ($-\omega^2 x$) is zero. At this same point, the velocity equation ($v = \pm\omega\sqrt{A^2-x^2}$) becomes maximum ($\pm\omega A$).
Important Q15: Assertion (A): The time period of a simple pendulum depends on the mass of the bob. Reason (R): A heavier mass will be pulled down faster by gravity.
Answer: D . The assertion is completely false. The time period of a simple pendulum ($T = 2\pi\sqrt{l/g}$) is entirely independent of the mass of the bob. Galileo famously proved that all masses fall at the same rate under gravity.
FAQ Section
Is Oscillations Chapter 13 important for JEE/NEET 2026?
Absolutely! It is incredibly important. Concepts from SHM are frequently combined with other topics (like Electrostatics or Magnetism) to create tricky, high-scoring questions in competitive exams.
Where can I download the Class 11 Physics NCERT PDF for Oscillations?
You can easily download the official, updated NCERT Physics textbook for free directly from the official NCERT website (ncert.nic.in) under the 'Publications' section.
What is the difference between Periodic and Oscillatory motion?
All oscillatory motion (back and forth) is periodic, but not all periodic motion is oscillatory. For example, a clock's hands move periodically, but they don't oscillate back and forth.
Why is the restoring force negative ($F = -kx$)?
The negative sign indicates that the force always acts in the opposite direction to the displacement, constantly trying to pull the object back to the equilibrium (mean) position.
What is the physical meaning of 'Angular Frequency' ($\omega$) in a straight-line SHM?
Even though the object moves in a straight line, SHM can be perfectly mapped to an imaginary particle moving in uniform circular motion. $\omega$ represents the speed at which this imaginary reference particle completes its circle ($2\pi$ radians)
More Class 11 Physics Chapters
Ch 1: Units and MeasurementsCh 2: Motion in a Straight LineCh 3: Motion in a PlaneCh 4: Laws of MotionCh 5: Work, Energy and PowerCh 6: System of Particles and Rotational MotionCh 7: GravitationCh 8: Mechanical Properties of SolidsCh 9: Mechanical Properties of FluidsCh 10: Thermal Properties of MatterCh 11: ThermodynamicsCh 12: Kinetic TheoryCh 13: OscillationsCh 14: Waves