**NCERT Solutions for Class 11 Physics Chapter 14 Oscillations**: Problems on motion along a straight line, motion in a plane, projectile motion, etc were discussed in the previous chapters. Solutions of NCERT class 11 physics chapter 14 oscillations explains problems on periodic and oscillatory motions. The motion which repeats after a certain interval of time is called periodic motion. For example, the motion of a planet around the sun, the motion of pendulum of a wall clock etc are periodic. To and fro periodic motion about a mean position is known as oscillatory motion. CBSE NCERT solutions for class 11 physics chapter 14 oscillations have questions on simple harmonic motion (SHM). SHM is the simplest form of oscillatory motion. In SHM the force on the oscillating body is directly proportional to the displacement about the mean position and is directed towards the mean position. NCERT solutions are an important tool to score well in the exams and also they are useful if you want to study other subjects of other classes as well. Concept studied in the chapter becomes easy to understand with the help of NCERT solutions for class 11 physics chapter 14 oscillations.

14.1 Introduction

14.2 Periodic and oscillatory motions

14.3 Simple harmonic motion

14.4 Simple harmonic motion and uniform circular motion

14.5 Velocity and acceleration in simple harmonic motion

14.6 Force law for simple harmonic motion

14.7 Energy in simple harmonic motion

14.8 Some systems executing simple harmonic motion

14.9 Damped simple harmonic motion

14.10 Forced oscillations and resonance

Other types of oscillatory motions that are discussed in the** **NCERT class 11 Physics Chapter 14 are damped oscillations and forced oscillations. In damped oscillation, as the name indicates the oscillation gets damped after an interval of time. And if the damping is small the motion will be approximately periodic, the position-time graph of such damping oscillation is shown below.

We have seen the oscillation of a pendulum, this oscillation will be damped unless an external force is applied to maintain the oscillation. Such maintained oscillation due to an external agency is called forced or driven oscillations.

**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 bank.

(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 motion is not periodic though it is to and fro.

(b) The motion is periodic.

(c) The motion is periodic.

(d) The motion is not 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.

(d) general vibrations of a polyatomic molecule about its equilibrium position.

(a) Periodic but not S.H.M.

(b) S.H.M.

(c) S.H.M.

(d) Periodic but not S.H.M.M [A polyatomic molecule has a number of natural frequencies, so its vibration is a superposition of SHM’s of a number of different frequencies. This is periodic but not SHM]

**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)?

The x-t plots for linear motion of a particle in Fig. 14.23 (b) and (d) represent periodic motion with both having a period of motion of two seconds.

**Q. 14.4 (a) **Which of the following functions of time represent

Since the above function is of form it represents SHM with a time period of

(b)

The two functions individually represent SHM but their superposition does not give rise to SHM but the motion will definitely be periodic with a period of

Here each individual functions are SHM. But superposition is not SHM. The function represents periodic motion but not SHM.

The given function is exponential and therefore does not represent periodic motion.

The given function does not represent periodic motion.

Velocity is zero. Force and acceleration are in the positive direction.

Velocity is zero. Acceleration and force are negative.

(c) at the mid-point of AB going towards ,

Velocity is negative that is towards A and its magnitude is maximum. Acceleration and force are zero.

(d) at away from going towards ,

Velocity is negative. Acceleration and force are also negative.

(e) at away from going towards , and

Velocity is positive. Acceleration and force are also positive.

Only the relation given in (c) represents simple harmonic motion as the acceleration is proportional in magnitude to the displacement from the midpoint and its direction is opposite to that of the displacement from the mean position.

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

at t = 0

at t = 0

Squaring and adding equation (i) and (ii) we get

Dividing equation (ii) by (i) we get

at t = 0

at t = 0

Squaring and adding equation (iii) and (iv) we get

Dividing equation (iii) by (iv) we get

Spring constant of the spring is given by

The time period of a spring attached to a body of mass m is given by

(i) the frequency of oscillations,

The frequency of oscillation of an object of mass m attached to a spring of spring constant k is given by

(ii) maximum acceleration of the mass, and

A body executing S.H.M experiences maximum acceleration at the extreme points

(F_{A} = Force experienced by body at displacement A from mean position)

(iii) the maximum speed of the mass.

Maximum speed occurs at the mean position and is given by

Amplitude is A = 0.02 m

Time period is

(a) At t = 0 the mass is at mean position i.e. at t = 0, x = 0

Here x is in metres and t is in seconds.

(b) at the maximum stretched position,

Amplitude is A = 0.02 m

Time period is

(b) At t = 0 the mass is at the maximum stretched position.

x(0) = A

Here x is in metres and t is in seconds.

(c) at the maximum compressed position.

Amplitude is A = 0.02 m

Time period is

(c) At t = 0 the mass is at the maximum compressed position.

x(0) = -A

Here x is in metres and t is in seconds.

The above functions differ only in the initial phase and not in amplitude or frequency.

(a) Let the required function be

Amplitude = 3 cm = 0.03 m

T = 2 s

Since initial position x(t) = 0,

As the sense of revolution is clock wise

Here x is in metres and t is in seconds.

(b)Let the required function be

Amplitude = 2 m

T = 4 s

Since initial position x(t) = -A,

As the sense of revolution is anti-clock wise

Here x is in metres and t is in seconds.

The initial position of the particle is x(0)

The radius of the circle i.e. the amplitude is 2 cm

The angular speed of the rotating particle is

Initial phase is

The reference circle for the given simple Harmonic motion is

The initial position of the particle is x(0)

The radius of the circle i.e. the amplitude is 1 cm

The angular speed of the rotating particle is

Initial phase is

The reference circle for the given simple Harmonic motion is

At t= 0

Reference circle is as follows

(d)

The initial position of the particle is x(0)

The radius of the circle i.e. the amplitude is 2 cm

The angular speed of the rotating particle is

Initial phase is

The reference circle for the given simple Harmonic motion is

**Q. 14.13 (a) **Figure 14.30

(b) is stretched by the same force F.

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

(a) Let us assume the maximum extension produced in the spring is x.

At maximum extension

(b) Let us assume the maximum extension produced in the spring is x. That is x/2 due to force towards left and x/2 due to force towards right

(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?

(b).(a) In Fig, (a) we have

F=-kx

ma=-kx

(b) In fig (b) the two equal masses will be executing SHM about their centre of mass. The time period of the system would be equal to a single object of same mass m attached to a spring of half the length of the given spring (or undergoing half the extension of the given spring while applied with the same force)

Spring constant of such a spring would be 2k

F=-2kx

ma=-2kx

Amplitude of SHM = 0.5 m

angular frequency is

If the equation of SHM is given by

The velocity would be given by

The maximum speed is therefore

The time period of a simple pendulum of length l executing S.H.M is given by

g_{e }= 9.8 m s^{-2}

g_{m} = 1.7 m s^{-2}

The time period of the pendulum on the surface of Earth is T_{e} = 3.5 s

The time period of the pendulum on the surface of the moon is T_{m}

**Q. 14.16 (a) ** Answer the following questions :

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

In case of spring, the spring constant is independent of the mass attached whereas in case of a pendulum k is proportional to m making k/m constant and thus the time period comes out to be independent of the mass of the body attached.

**Q. 14.16 (b) **Answer the following questions :

In reaching the result we have assumed sin(x/l)=x/l. This assumption is only true for very small values of x . Therefore it is obvious that once x takes larger values we will have deviations from the above-mentioned value.

**Q. 14.16 (c) **Answer the following questions :

The watch must be using an electrical circuit or a spring system to tell the time and therefore free falling would not affect the time his watch predicts.

**Q. 14.16 (d) **Answer the following questions :

While free falling the effective value of g inside the cabin will be zero and therefore the frequency of oscillation of a simple pendulum would be zero i.e. it would not vibrate at all because of the absence of a restoring force.

Acceleration due to gravity = g (in downwards direction)

Centripetal acceleration due to the circular movement of the car = a_{c}

(in the horizontal direction)

Effective acceleration is

The time period is T'

Let the cork be displaced by a small distance x in downwards direction from its equilibrium position where it is floating.

The extra volume of fluid displaced by the cork is Ax

Taking the downwards direction as positive we have

Comparing with a=-kx we have

Let the height of each mercury column be h.

The total length of mercury in both the columns = 2h.

Let the cross-sectional area of the mercury column be A.

Let the density of mercury be

When either of the mercury columns dips by a distance x, the total difference between the two columns becomes 2x.

Weight of this difference is

This weight drives the rest of the entire column to the original mean position.

Let the acceleration of the column be a Since the force is restoring

** which is the equation of a body executing S.H.M**

The time period of the oscillation would be

**NCERT solutions for class 11 physics chapter 14 oscillations additional exercise:**

Let the initial volume and pressure of the chamber be V and P.

Let the ball be pressed by a distance x.

This will change the volume by an amount ax.

Let the change in pressure be

Let the Bulk's modulus of air be K.

This pressure variation would try to restore the position of the ball.

Since force is restoring in nature displacement and acceleration due to the force would be in different directions.

The above is the equation of a body executing S.H.M.

The time period of the oscillation would be

Mass of automobile (m) = 3000 kg

There are a total of four springs.

Compression in each spring, x = 15 cm = 0.15 m

Let the spring constant of each spring be k

The amplitude of oscillation decreases by 50 % in one oscillation i.e. in one time period.

For damping factor b we have

x=x_{0}/2

t=0.77s

m=750 kg

Let the equation of oscillation be given by

Velocity would be given as

Kinetic energy at an instant is given by

Time Period is given by

The Average Kinetic Energy would be given as follows

The potential energy at an instant T is given by

The Average Potential Energy would be given by

We can see K_{av} = U_{av}

Moment of Inertia of the disc about the axis passing through its centre and perpendicular to it is

The period of Torsional oscillations would be

A = 5 cm = 0.05 m

T = 0.2 s

At displacement x acceleration is

At displacement x velocity is

(a)At displacement 5 cm

A = 5 cm = 0.05 m

T = 0.2 s

At displacement x acceleration is

At displacement x velocity is

(a)At displacement 3 cm

A = 5 cm = 0.05 m

T = 0.2 s

At displacement x acceleration is

At displacement x velocity is

(a)At displacement 0 cm

At the maximum extension of spring, the entire energy of the system would be stored as the potential energy of the spring.

Let the amplitude be A

The angular frequency of a spring-mass system is always equal to

Therefore

On an average 6.67 % of questions from oscillation and waves are asked for JEE Mains. Most of the previous JEE mains questions from oscillation asked are from topics SHM and simple pendulum. For NEET exam 2 questions are expected from oscillation. The CBSE NCERT solutions for class 11 physics chapter 14 oscillations will help to score well in class 11 and competitive exams.

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