NCERT Solutions for Class 11 Physics Chapter 13 Oscillations – Free PDF Download

Q: 14.1 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 centre of mass (d) An arrow released from a bow

(a) The swimmer's motion is not periodic. The motion of the swimmer between the banks of a river is back and forth. However, it does not have a definite period. This is because the time taken by the swimmer during his back and forth journey may not be the same. (b) The motion of a freely-suspended magnet, if displaced from its N-S direction and released, is periodic. This is because the magnet oscillates about its position with a definite period of time. (c) When a hydrogen molecule rotates about its centre of mass, it comes to the same position again and again, after an equal interval of time. Such motion is periodic. (d) An arrow released from a bow moves only in the forward direction. It does not come backward. Hence this motion is not a periodic.

Q: 14.2 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.

(a) During its rotation about its axis, earth comes to the same position again and again in equal intervals of time. Hence it is a periodic motion. However, this motion is not simple harmonic, as earth does not have to and fro motion about its axis. (b) An oscillating mercury column in a U-tube is simple harmonic. This is because the mercury moves to and fro on the same path, about the fixed position, with a certain period of time. (c) The ball moves to and fro about the lowermost point of the bowl when released. Also the ball comes back to its initial position in the same period of time, again and again. Hence, its motion is periodic as well as simple harmonic. (d) A polyatomic molecule has many natural frequencies of oscillations. Their vibration is the superposition of individual simple harmonic motion of a number of different molecules. Hence, it is not simple harmonic, it is periodic.

Q: 14.3 Fig. 14.23 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)?

(a) It is not a periodic motion. It represents a unidirectional, linear uniform motion. There is no repetition of motion (b) In this case, the motion of the particle repeats itself after 2 s. Hence, it is a periodic motion, having a period of 2 s (c) It is not a periodic motion. This is because the particle repeats the motion in one position only. For a periodic motion, the entire motion of the particle must be repeated in equal intervals of time (d) In this case, the motion of the particle repeats itself after 2 s. Hence, it is periodic motion, having a period of 2 s

Q: 14.4 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 ( ω is any positive constant): (a) Sin ω t – cos ω t (b) Sin3 ω t (c) 3 cos ( π /4 – 2 ω t) (d) Cos ω t + cos 3 ω t + cos 5 ω t (e) Exp (– ω 2t2) (f) 1 + ω t + ω 2t2

(a) The given function is sin ω t – cos ω t = √21√2sin?ωt-1√2cos?ωt = √21√2sin?ωt×cos?π4-1√2cos?ωt×sin?π4 = √2sinωt-π4 ) This function represents SHM as it can be written in the form: a sin ( ω t + θ ) Its period is 2πω . It is periodic, but not SHM (b) Sin3 ω t = 123sin?ωt-sin?3ωt The terms sin?ω t and sin?3ω t individually represents simple harmonic motion. However, the superposition of two SHM is periodic but not simple harmonic. (c) The given function is 3 cos ( π /4 – 2 ω t) = -3cos(2 ω t - π /4) This function represents simple harmonic motion because it can be written in the form: a cos( ωt+θ ), its time period is 2π2ω = πω . Periodic but not SHM (d) The given function is cos ω t + cos 3 ω t + cos 5 ω t – Each individual cosine function represents SHM, however the superposition of three SHM is periodic but not simple harmonic. (e) The given function exp (– ω 2t2) is an exponential function. Exponential functions do not repeat themselves, hence non-periodic motion. (f) The given function 1 + ω t + ω 2t2 is non-periodic.

Q: 14.5 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 (f) At 4 cm away from B going towards A

a) The given situation is shown in the figure. Points A and B are the two end points, with AB = 10 cm. O is the midpoint of the path. A particle is in linear simple harmonic motion between the end points. At the extreme point A, the particle is at rest momentarily. Hence, its velocity is zero at this point. Its acceleration is positive as it is directed along AO. Force is also positive in this case as the particle is directed rightward. (b) At the extreme point B, the particle is at rest momentarily. Hence, its velocity is zero at this point. Its acceleration is negative in this case as it is directed along B. Force is also negative as it is directed leftward. (c) The particle is executing a simple harmonic motion. O is the mean position of the particle. Its velocity at the mean position O is the maximum. The value for velocity is negative as the particle is directed leftward. The acceleration and force of a particle executing SHM is zero at the mean position. (d) The particle is moving towards point O from the end B. This direction of motion is opposite to the conventional positive direction, which is from A to B. Hence, the particle's velocity and acceleration, and the force on it are all negative. (e) The particle is moving towards point O from the end A. This direction of motion is from A to B, which is the conventional positive direction. Hence, the values for velocity, acceleration and force are all positive. (f) The particle is moving towards point O from the end B. This direction of motion is opposite to the conventional positive direction, which is from A to B. Hence, the particle's velocity and acceleration, and the force on it are all negative.

Updated on Aug 12, 2025 12:02 IST

By Pallavi Pathak, Assistant Manager Content

Oscillations Class 11 Physics covers the motion of the object to and fro about a mean position, similar to the pendulum of a wall clock. Such motion is called the oscillatory motion. Other examples of the oscillatory motion include the piston in a steam engine going back and forth, and a boat tossing up and down in a river.
Oscillations Class 11 NCERT Solutions are important for students as the concept of the oscillatory motion is basic to physics, and its understanding is important for many physical phenomena. The key concepts of this chapter include the frequency, period, amplitude, displacement, and phase. The experts at Shiksha created these solutions to help students understand the concepts clearly and achieve great success in their Class 11 exams, CBSE Board, and other entrance tests like NEET and JEE Main.
Students can access the comprehensive NCERT solutions for Class 11 and Class 12 of three subjects - Physics, Chemistry, and Maths at NCERT Solutions Class 11 and 12.

Table of content

NCERT Class 11 Physics Oscillation: Weightage, Key Topics
Important Formulas of Oscillations Class 11
NCERT Physics Class11th Solution for Oscillations PDF Download
Chapter 13 Oscillations Important Formulas & Concepts
NCERT Physics Class11th Oscillations Solutions

NCERT Class 11 Physics Oscillation: Weightage, Key Topics

Important topics of Oscillations Class 11 include time period, amplitude, and frequency of oscillations, Simple Harmonic Motion (SHM), Energy in SHM, law of simple pendulum, wave motion, including types of waves, damped and forced oscillations, standing waves, superposition of waves, and Doppler effect. See below the topics covered in this chapter:

Exercise	Topics Covered
13.1	Introduction
13.2	Periodic and Oscillatory Motions
13.3	Simple Harmonic Motion
13.4	Simple Harmonic Motion and Uniform Circular Motion
13.5	Velocity and Acceleration in Simple Harmonic Motion
13.6	Force Law for Simple Harmonic Motion
13.7	Energy in Simple Harmonic Motion
13.8	The Simple Pendulum

Oscillations Class 11 Weightage in NEET, JEE Main exam

Exam	Number of Questions	Weightage
JEE Main	2-3 questions	9%
NEET	1-3 questions	3% to 7%

$PE = \frac{1}{2} m \omega^2 x^2$

Important Formulas of Oscillations Class 11

The following are important formulas of this chapter:

Physical Quantity	Symbol	Formula
Frequency	v	$T = \frac{1}{v}$
Angular Frequency	$ω$	$ω = 2 π ν$
Force Constant	k	$\begin{matrix} Simple harmonic motion \\ F = - k x \end{matrix}$

NCERT Physics Class11th Solution for Oscillations PDF Download

It is recommended for students to download the oscillations class 11 NCERT PDF from the link given below. It offers accurate and reliable study material for the school exam, CBSE Board, and competitive exam preparation.

Download Here: NCERT Solution for Class XI Physics Chapter Oscillation PDF

Chapter 13 Oscillations Important Formulas & Concepts

Important Formulae of Oscillations for CBSE & Competitive Exams

Equation of SHM:
$x(t) = A \sin(\omega t + \phi)$
Where $A$ is amplitude, $\omega$ is angular frequency, and $\phi$ is phase constant
Velocity in SHM:
$v(t) = \omega \sqrt{A^2 - x^2}$
Acceleration in SHM:
$a(t) = -\omega^2 x$
Time Period of a Spring-Mass System:
$T = 2\pi \sqrt{\frac{m}{k}}$
Time Period of a Simple Pendulum:
$T = 2\pi \sqrt{\frac{l}{g}}$
Total Energy in SHM:
$E = \frac{1}{2}kA^2 = \frac{1}{2}m\omega^2 A^2$
Angular Frequency:
$\omega = 2\pi f = \sqrt{\frac{k}{m}}$
Energy Distribution:
- Kinetic Energy: $KE = \frac{1}{2} m \omega^2 (A^2 - x^2)$
- Potential Energy: $P E = \frac{1}{2} m ω^{2} x^{2}$

NCERT Physics Class11th Oscillations Solutions

Q.14.1 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 centre of mass

(d) An arrow released from a bow

Ans.14.1

(a) The swimmer’s motion is not periodic. The motion of the swimmer between the banks of a river is back and forth. However, it does not have a definite period. This is because the time taken by the swimmer during his back and forth journey may not be the same.

(b) The motion of a freely-suspended magnet, if displaced from its N-S direction and released, is periodic. This is because the magnet oscillates about its position with a definite period of time.

(c) When a hydrogen molecule rotates about its centre of mass, it comes to the same position again and again, after an equal interval of time. Such motion is periodic.

(d) An arrow released from a bow moves only in the forward direction. It does not come backward. Hence this motion is not a periodic.

Q.14.2 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.

Ans.14.2

(a) During its rotation about its axis, earth comes to the same position again and again in equal intervals of time. Hence it is a periodic motion. However, this motion is not simple harmonic, as earth does not have to and fro motion about its axis.

(b) An oscillating mercury column in a U-tube is simple harmonic. This is because the mercury moves to and fro on the same path, about the fixed position, with a certain period of time.

(c) The ball moves to and fro about the lowermost point of the bowl when released. Also the ball comes back to its initial position in the same period of time, again and again. Hence, its motion is periodic as well as simple harmonic.

(d) A polyatomic molecule has many natural frequencies of oscillations. Their vibration is the superposition of individual simple harmonic motion of a number of different molecules. Hence, it is not simple harmonic, it is periodic.

Q.14.3 Fig. 14.23 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)?

Ans.14.3

(a) It is not a periodic motion. It represents a unidirectional, linear uniform motion. There is no repetition of motion

(b) In this case, the motion of the particle repeats itself after 2 s. Hence, it is a periodic motion, having a period of 2 s

(c) It is not a periodic motion. This is because the particle repeats the motion in one position only. For a periodic motion, the entire motion of the particle must be repeated in equal intervals of time

(d) In this case, the motion of the particle repeats itself after 2 s. Hence, it is periodic motion, having a period of 2 s

Q.14.4 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 ( $ω$ is any positive constant):

(a) Sin $ω$ t – cos $ω$ t

(b) Sin³ $ω$ t

(c) 3 cos ( $π$ /4 – 2 $ω$ t)

(d) Cos $ω$ t + cos 3 $ω$ t + cos 5 $ω$ t

(e) Exp (– $ω$ 2t2)

(f) 1 + $ω$ t + $ω$ ²t²

Ans.14.4

(a) The given function is sin $ω$ t – cos $ω$ t

= $\sqrt 2 [\frac{1}{\sqrt 2} \sin ω t - \frac{1}{\sqrt 2} \cos ω t]$ = $\sqrt 2 [\frac{1}{\sqrt 2} \sin ω t \times \cos \frac{π}{4} - \frac{1}{\sqrt 2} \cos ω t \times \sin \frac{π}{4}]$

= $\sqrt 2 s i n [ω t - \frac{π}{4}]$ )

This function represents SHM as it can be written in the form: a sin ( $ω$ t + $θ$ )

Its period is $\frac{2 π}{ω}$ . It is periodic, but not SHM

(b) Sin³ $ω$ t = $\frac{1}{2} [3 \sin ω t - \sin 3 ω t]$

The terms $\sin ω$ t and $\sin 3 ω$ t individually represents simple harmonic motion. However, the superposition of two SHM is periodic but not simple harmonic.

(c) The given function is 3 cos ( $π$ /4 – 2 $ω$ t) = -3cos(2 $ω$ t - $π$ /4)

This function represents simple harmonic motion because it can be written in the form: a cos( $ω t + θ$ ), its time period is $\frac{2 π}{2 ω}$ = $\frac{π}{ω}$ . Periodic but not SHM

(d) The given function is cos $ω$ t + cos 3 $ω$ t + cos 5 $ω$ t – Each individual cosine function represents SHM, however the superposition of three SHM is periodic but not simple harmonic.

(e) The given function exp (– $ω$ ²t²) is an exponential function. Exponential functions do not repeat themselves, hence non-periodic motion.

(f) The given function 1 + $ω$ t + $ω$ ²t² is non-periodic.

Commonly asked questions

14.1 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 centre of mass

(d) An arrow released from a bow

14.2 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.

14.3 Fig. 14.23 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)?

14.4 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 ( $ω$ is any positive constant):

(a) Sin $ω$ t – cos $ω$ t

(b) Sin³ $ω$ t

(c) 3 cos ( $π$ /4 – 2 $ω$ t)

(d) Cos $ω$ t + cos 3 $ω$ t + cos 5 $ω$ t

(e) Exp (– $ω$ 2t2)

(f) 1 + $ω$ t + $ω$ ²t²

14.5 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

(f) At 4 cm away from B going towards A

14.6 Which of the following relationships between the acceleration a and the displacement x of a particle involve simple harmonic motion?

(a) A = 0.7x

(b) A = –200x²

(c) A = –10x

(d) A = 100x³

14.7 The motion of a particle executing simple harmonic motion is described by the displacement function,

x(t) = A cos ( $ω$ t + $φ$ ).

If the initial (t = 0) position of the particle is 1 cm and its initial velocity is $ω$ cm/s, what are its amplitude and initial phase angle ? The angular frequency of the particle is $π$ s^–1. If instead of the cosine function, we choose the sine function to describe the SHM : x = B sin ( $ω$ t + $α$ ), what are the amplitude and initial phase of the particle with the above initial conditions.

14.8 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?

14.9 A spring having with a spring constant 1200 N m–1 is mounted on a horizontal table as shown in Fig. 14.24. A mass of 3 kg is attached to the free end of the spring. The mass is then pulled sideways to a distance of 2.0 cm and released.

Determine

(i) The frequency of oscillations

(ii) Maximum acceleration of the mass

(iii) The maximum speed of the mass

14.10 In Exercise 14.9, let us take the position of mass when the spring is unstretched as x = 0, and the direction from left to right as the positive direction of x-axis. Give x as a function of time t for the oscillating mass if at the moment we start the stopwatch (t = 0), the mass is

(a) At the mean position

(b) At the maximum stretched position

(c) At the maximum compressed position

In what way do these functions for SHM differ from each other, in frequency, in amplitude or the initial phase?

14.11 Figures 14.25 correspond to two circular motions. The radius of the circle, the period of revolution, the initial position, and the sense of revolution (i.e. clockwise or anti-clockwise) are indicated on each figure.

Obtain the corresponding simple harmonic motions of the x-projection of the radius vector of the revolving particle P, in each case.

14.12 Plot the corresponding reference circle for each of the following simple harmonic motions. Indicate the initial (t =0) position of the particle, the radius of the circle, and the angular speed of the rotating particle. For simplicity, the sense of rotation may be fixed to be anticlockwise in every case: (x is in cm and t is in s).

(a) X = –2 sin (3t + $π$ /3)

(b) X = cos ( $π$ 6 – t)

(c) X = 3 sin (2 $π$ t + $π$ /4)

(d) X = 2 cos $π$ t

14.13 Figure 14.26 (a) shows a spring of force constant k clamped rigidly at one end and a mass m attached to its free end. A force F applied at the free end stretches the spring. Figure 14.26 (b) shows the same spring with both ends free and attached to a mass m at either end. Each end of the spring in Fig. 14.26(b) is stretched by the same force F.

(a) What is the maximum extension of the spring in the two cases ?

(b) If the mass in Fig. (a) and the two masses in Fig. (b) are released, what is the period of oscillation in each case ?

14.14 The piston in the cylinder head of a locomotive has a stroke (twice the amplitude) of 1.0 m. If the piston moves with simple harmonic motion with an angular frequency of 200 rad/min, what is its maximum speed ?

14.15 The acceleration due to gravity on the surface of moon is 1.7 m s^–2. What is the time period of a simple pendulum on the surface of moon if its time period on the surface of earth is 3.5 s ? (g on the surface of earth is 9.8 m s^–2)

14.16 Answer the following questions :

(a) Time period of a particle in SHM depends on the force constant k and mass m of the particle

T = 2 $π \sqrt{\frac{m}{k}} .$ A simple pendulum executes SHM approximately. Why then is the time period of a pendulum independent of the mass of the pendulum?

(b) The motion of a simple pendulum is approximately simple harmonic for small angle oscillations. For larger angles of oscillation, a more involved analysis shows that T is greater than 2 $π \sqrt{\frac{l}{g}}$ . Think of a qualitative argument to appreciate this result.

(c) A man with a wristwatch on his hand falls from the top of a tower. Does the watch give correct time during the free fall ?

(d) What is the frequency of oscillation of a simple pendulum mounted in a cabin that is freely falling under gravity?

14.17 A simple pendulum of length l and having a bob of mass M is suspended in a car. The car is moving on a circular track of radius R with a uniform speed v. If the pendulum makes small oscillations in a radial direction about its equilibrium position, what will be its time period?

.14.18 A cylindrical piece of cork of density of base area A and height h floats in a liquid of density $ρ_{l}$ .The cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically with a period

T = 2 $π \sqrt{\frac{h ρ}{ρ_{l} g}}$

where $ρ$ is the density of cork. (Ignore damping due to viscosity of the liquid).

14.19 One end of a U-tube containing mercury is connected to a suction pump and the other end to atmosphere. A small pressure difference is maintained between the two columns. Show that, when the suction pump is removed, the column of mercury in the U-tube executes simple harmonic motion.

14.20 An air chamber of volume V has a neck area of cross section a into which a ball of mass m just fits and can move up and down without any friction (Fig.14.27). Show that when the ball is pressed down a little and released , it executes SHM. Obtain an expression for the time period of oscillations assuming pressure-volume variations of air to be isothermal [see Fig. 14.27].

14.21 You are riding in an automobile of mass 3000 kg. Assuming that you are examining the oscillation characteristics of its suspension system. The suspension sags 15 cm when the entire automobile is placed on it. Also, the amplitude of oscillation decreases by 50% during one complete oscillation.

Estimate the values of

(a) The spring constant k and

(b) The damping constant b for the spring and shock absorber system of one wheel, assuming that each wheel supports 750 kg.

14.22 Show that for a particle in linear SHM the average kinetic energy over a period of oscillation equals the average potential energy over the same period.

14.23 A circular disc of mass 10 kg is suspended by a wire attached to its centre. The wire is twisted by rotating the disc and released. The period of torsional oscillations is found to be 1.5 s. The radius of the disc is 15 cm. Determine the torsional spring constant of the wire. (Torsional spring constant $α$ is defined by the relation J = – $α θ$ , where J is the restoring couple and $θ$ the angle of twist).

14.24 A body describes simple harmonic motion with an amplitude of 5 cm and a period of 0.2 s. Find the acceleration and velocity of the body when the displacement is (a) 5 cm (b) 3 cm (c) 0 cm.

14.25 A mass attached to a spring is free to oscillate, with angular velocity $ω$ , in a horizontal plane without friction or damping. It is pulled to a distance $x_{0}$ and pushed towards the centre with a velocity $v_{0}$ at time t = 0. Determine the amplitude of the resulting oscillations in terms of the parameters $ω, x_{0}$ and $v_{0}$ .