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Physics · Class 11 Science

Ch 13Oscillations — Class 11 Physics, concept-first.

You have already studied several kinds of motion: a car moving along a straight road (rectilinear motion), a cricket ball thrown through the air (projectile motion). These are non-repetitive — the object never returns to exactly the same state in the same way.

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Key concepts

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Frequency And Period

Imagine a child on a swing, going back and forth. Each full cycle — forward, backward, back to the start — takes a certain amount of time. That time is the period.

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

13.1

Introduction

You have already studied several kinds of motion: a car moving along a straight road (rectilinear motion), a cricket ball thrown through the air (projectile motion).

13.2

Periodic and Oscillatory Motions

Consider three everyday situations: an insect climbing up a ramp and falling back down, over and over; a child climbing a step and coming back down, again and again; and a ball bouncing between your p…

13.2.1

Period and Frequency

When you push a child on a swing, the swing goes back and forth, returning to where it started after a fixed interval. That fixed interval is the period.

+Worked Examplesi1 question
  1. Example 13.1On an average, a human heart is found to beat 75 times in a minute. Calculate its frequency and period.Preview
13.2.2

Displacement

When we talk about displacement in the context of oscillations, we are not just talking about how far something has moved from where it started.

+Worked Examplesi1 question
  1. Example 13.2Which of the following functions of time represent (a) periodic and (b) non-periodic motion? Give the period for each case of periodic motio…Preview
13.3

Simple Harmonic Motion

Simple harmonic motion (SHM) is the most fundamental kind of oscillation. It occurs when the restoring force on a particle is directly proportional to its displacement from a fixed equilibrium positio…

+Worked Examplesi1 question
  1. Example 13.3Which of the following functions of time represent (a) simple harmonic motion and (b) periodic but not simple harmonic? Give the period for…Preview
13.4

Simple Harmonic Motion and Uniform Circular Motion

Simple harmonic motion and uniform circular motion might seem like completely different phenomena — one is a back-and-forth oscillation along a straight line, the other is steady rotation around a cir…

+Worked Examplesi1 question
  1. Example 13.4The figure given below depicts two circular motions. The radius of the circle, the period of revolution, the initial position and the sense…Preview
13.5

Velocity and Acceleration in Simple Harmonic Motion

When a particle executes simple harmonic motion, its position changes sinusoidally with time. From that single fact, we can derive everything about how fast it moves and how it accelerates.

+Worked Examplesi1 question
  1. Example 13.5A body oscillates with SHM according to the equation (in SI units), $x = 5\cos\left[2\pi t + \pi/4\right]$. At $t = 1.5$ s, calculate the (a…Preview
13.6

Force Law for Simple Harmonic Motion

Simple harmonic motion is not just a pattern of motion — it is defined by a specific relationship between the force acting on a particle and its displacement from equilibrium.

+Worked Examplesi1 question
  1. Example 13.6Two identical springs of spring constant $k$ are attached to a block of mass $m$ and to fixed supports as shown in Fig. 13.14. Show that whe…Preview
13.7

Energy in Simple Harmonic Motion

When a particle oscillates in simple harmonic motion, it continuously exchanges energy between two forms: kinetic energy (due to its motion) and potential energy (due to its displacement from equilibr…

+Worked Examplesi1 question
  1. Example 13.7A block whose mass is 1 kg is fastened to a spring. The spring has a spring constant of $50\ \text{N m}^{-1}$. The block is pulled to a dist…Preview
13.8

The Simple Pendulum

A simple pendulum is one of the most familiar examples of oscillatory motion. It consists of a point mass (called the bob) suspended from a fixed point by a massless, inextensible string.

+Worked Examplesi1 question
  1. Example 13.8What is the length of a simple pendulum, which ticks seconds?Preview

Summary

- Simple harmonic motion (SHM) occurs when the restoring force is proportional to displacement and opposite in direction: . The equation of motion is , where is the angular frequency.

Points to Ponder

1. The period is the shortest time after which the motion exactly repeats itself. Because of this, the motion also repeats after any integer multiple , where

Exercises

+Show 18 questions18 questions
  1. 13.1Which of the following examples represent periodic motion? (a) A swimmer completing one (return) trip from one bank of a river to the other…Free
  2. 13.2Which of the following examples represent (nearly) simple harmonic motion and which represent periodic but not simple harmonic motion? (a) t…Free
  3. 13.3Four displacement-time ($x$-$t$) plots for the linear motion of a particle are described below. In each plot $x$ is on the vertical axis and…Free
  4. 13.4Which of the following functions of time represent (a) simple harmonic, (b) periodic but not simple harmonic, and (c) non-periodic motion? G…Preview
  5. 13.5A 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 dire…Preview
  6. 13.6Which of the following relationships between the acceleration $a$ and the displacement $x$ of a particle involve simple harmonic motion? (a)…Preview
  7. 13.7The motion of a particle executing simple harmonic motion is described by the displacement function, $x(t) = A\cos(\omega t + \phi)$. If the…Preview
  8. 13.8A 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 displace…Preview
  9. 13.9A spring having with a spring constant $1200\ \text{N m}^{-1}$ is mounted on a horizontal table as shown in Fig. 13.19. A mass of 3 kg is at…Preview
  10. 13.10In Exercise 13.9, let us take the position of mass when the spring is unstretched as $x = 0$, and the direction from left to right as the po…Preview
  11. 13.11Two particles each move uniformly on a circle lying in a plane, the centre of each circle at the origin of its own $x$ and $y$ axes. For eac…Preview
  12. 13.12Plot the corresponding reference circle for each of the following simple harmonic motions. Indicate the initial ($t = 0$) position of the pa…Preview
  13. 13.13Figure 13.21(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…Preview
  14. 13.14The 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…Preview
  15. 13.15The acceleration due to gravity on the surface of moon is $1.7\ \text{m s}^{-2}$. What is the time period of a simple pendulum on the surfac…Preview
  16. 13.16A 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…Preview
  17. 13.17A cylindrical piece of cork of density $\rho$ of base area $A$ and height $h$ floats in a liquid of density $\rho_l$. The cork is depressed…Preview
  18. 13.18One end of a U-tube containing mercury is connected to a suction pump and the other end to atmosphere. A small pressure difference is mainta…Preview

NCERT Exemplar

Higher-order thinking problems from the NCERT Exemplar.

+Show 40 questions40 questions
  1. Q1The displacement of a particle is represented by the equation $y = 3\cos\left(\dfrac{\pi}{4} - 2\omega t\right)$. The motion of the particle…Free
  2. Q2The displacement of a particle is represented by the equation $y = \sin^{3}\omega t$. The motion is (a) non-periodic. (b) periodic but not s…Free
  3. Q3The relation between acceleration and displacement of four particles are given below: (a) $a_x = +2x$. (b) $a_x = +2x^{2}$. (c) $a_x = -2x^{…Free
  4. Q4Motion of an oscillating liquid column in a U-tube is (a) periodic but not simple harmonic. (b) non-periodic. (c) simple harmonic and time p…Preview
  5. Q5A particle is acted simultaneously by mutually perpendicular simple hormonic motions $x = a\cos\omega t$ and $y = a\sin\omega t$. The trajec…Preview
  6. Q6The displacement of a particle varies with time according to the relation $y = a\sin\omega t + b\cos\omega t$. (a) The motion is oscillatory…Preview
  7. Q7Four pendulums A, B, C and D hang side by side in a row from a common horizontal elastic (flexible) support. Their lengths are related as fo…Preview
  8. Q8A particle P moves uniformly around a circle of radius $B$ centred at the origin O, with the $x$-axis horizontal (positive to the right) and…Preview
  9. Q9The equation of motion of a particle is $x = a\cos(\alpha t)^{2}$. The motion is (a) periodic but not oscillatory. (b) periodic and oscillat…Preview
  10. Q10A particle executing S.H.M. has a maximum speed of 30 cm/s and a maximum acceleration of 60 cm/s$^2$. The period of oscillation is (a) $\pi$…Preview
  11. Q11When a mass $m$ is connected individually to two springs $S_1$ and $S_2$, the oscillation frequencies are $\nu_1$ and $\nu_2$ respectively.…Preview
  12. Q12The rotation of earth about its axis is (Note: more than one of the given options may be correct.) (a) periodic motion. (b) simple harmonic…Preview
  13. Q13Motion of a ball bearing inside a smooth curved bowl, when released from a point slightly above the lower point is (Note: more than one of t…Preview
  14. Q14The displacement–time curve of a particle executing S.H.M. is a smooth sinusoid of period $4\ \text{s}$. The displacement is positive and ne…Preview
  15. Q15Which of the following statements is/are true for a simple harmonic oscillator? (Note: more than one of the given options may be correct.) (…Preview
  16. Q16The displacement–time graph of a particle executing S.H.M. is a cosine curve of period $T$: the displacement is at its positive maximum at $…Preview
  17. Q17A body is performing S.H.M. Then its (Note: more than one of the given options may be correct.) (a) average total energy per cycle is equal…Preview
  18. Q18A particle is in linear simple harmonic motion between two extreme points A and B lying on a straight line $10\ \text{cm}$ apart. The mean (…Preview
  19. Q19The displacement–time graph of a particle executing S.H.M. is a sinusoidal curve on which eight points are marked in time order: A, B, C, D,…Preview
  20. Q20Two identical springs, each of spring constant $K$, are attached to a block of mass $m$ resting on a smooth horizontal surface. One spring c…Preview
  21. Q21What are the two basic characteristics of a simple harmonic motion?Preview
  22. Q22When will the motion of a simple pendulum be simple harmonic?Preview
  23. Q23What is the ratio of maxmimum acceleration to the maximum velocity of a simple harmonic oscillator?Preview
  24. Q24What is the ratio between the distance travelled by the oscillator in one time period and amplitude?Preview
  25. Q25A reference particle P moves anticlockwise at constant speed around a circle of radius $R$ centred at the origin O, with the $x$-axis horizo…Preview
  26. Q26Show that for a particle executing S.H.M, velocity and displacement have a phase difference of $\pi/2$.Preview
  27. Q27Draw a graph to show the variation of P.E., K.E. and total energy of a simple harmonic oscillator with displacement.Preview
  28. Q28The length of a second's pendulum on the surface of Earth is 1m. What will be the length of a second's pendulum on the moon?Preview
  29. Q29A light spring of force constant $k$ hangs vertically from a rigid ceiling. Its lower end is tied to a light, frictionless movable pulley. A…Preview
  30. Q30Show that the motion of a particle represented by $y = \sin\omega t - \cos\omega t$ is simple harmonic with a period of $2\pi/\omega$.Preview
  31. Q31Find the displacement of a simple harmonic oscillator at which its P.E. is half of the maximum energy of the oscillator.Preview
  32. Q32A body of mass $m$ is situated in a potential field $U(x) = U_0(1 - \cos\alpha x)$ when $U_0$ and $\alpha$ are constants. Find the time peri…Preview
  33. Q33A mass of 2 kg is attached to the spring of spring constant $50\ \mathrm{Nm^{-1}}$. The block is pulled to a distance of 5cm from its equili…Preview
  34. Q34Consider a pair of identical pendulums, which oscillate with equal amplitude independently such that when one pendulum is at its extreme pos…Preview
  35. Q35A person normally weighing 50 kg stands on a massless platform which oscillates up and down harmonically at a frequency of $2.0\ \mathrm{s^{…Preview
  36. Q36A body of mass $m$ is attached to one end of a massless spring which is suspended vertically from a fixed point. The mass is held in hand so…Preview
  37. Q37A cylindrical log of wood of height $h$ and area of cross-section $A$ floats in water. It is pressed and then released. Show that the log wo…Preview
  38. Q38One end of a V-tube containing mercury is connected to a suction pump and the other end to atmosphere. The two arms of the tube are inclined…Preview
  39. Q39A tunnel is dug through the centre of the Earth. Show that a body of mass '$m$' when dropped from rest from one end of the tunnel will execu…Preview
  40. Q40A simple pendulum of time period $1\ \text{s}$ and length $l$ is hung from a fixed support at O. The pendulum swings in a vertical plane; wh…Preview