Course Content
Class 11 Physics Chapter 1 Physical World
Section Name Topic Name 1 Physical World 1.1 What is physics? 1.2 Scope and excitement of physics 1.3 Physics, technology and society 1.4 Fundamental forces in nature 1.5 Nature of physical laws
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Class 11 Physics Chapter 4 Motion In A Plane
4 Motion in a plane 4.1 Introduction 4.2 Scalars and vectors 4.3 Multiplication of vectors by real numbers 4.4 Addition and subtraction of vectors – graphical method 4.5 Resolution of vectors 4.6 Vector addition – analytical method 4.7 Motion in a plane 4.8 Motion in a plane with constant acceleration 4.9 Relative velocity in two dimensions 4.10 Projectile motion 4.11 Uniform circular motion
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Class 11 Physics Chapter 5 Laws of motion
Section Name Topic Name 5 Laws of motion 5.1 Introduction 5.2 Aristotle’s fallacy 5.3 The law of inertia 5.4 Newton’s first law of motion 5.5 Newton’s second law of motion 5.6 Newton’s third law of motion 5.7 Conservation of momentum 5.8 Equilibrium of a particle 5.9 Common forces in mechanics 5.10 Circular motion 5.11 Solving problems in mechanics
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Class 11 Physics Chapter 6 Work Energy and Power
Section Name Topic Name 6 Work Energy and power 6.1 Introduction 6.2 Notions of work and kinetic energy : The work-energy theorem 6.3 Work 6.4 Kinetic energy 6.5 Work done by a variable force 6.6 The work-energy theorem for a variable force 6.7 The concept of potential energy 6.8 The conservation of mechanical energy 6.9 The potential energy of a spring 6.10 Various forms of energy : the law of conservation of energy 6.11 Power 6.12 Collisions
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Class 11 Physics Chapter 7 Rotation motion
Topics Introduction Centre of mass Motion of COM Linear Momentum of System of Particles Vector Product Angular velocity Torque & Angular Momentum Conservation of Angular Momentum Equilibrium of Rigid Body Centre of Gravity Moment of Inertia Theorem of perpendicular axis Theorem of parallel axis Moment of Inertia of Objects Kinematics of Rotational Motion about a Fixed Axis Dynamics of Rotational Motion about a Fixed Axis Angular Momentum In Case of Rotation about a Fixed Axis Rolling motion
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Class 11 Physics Chapter 9 mechanics properties of solid
Section Name Topic Name 9 Mechanical Properties Of Solids 9.1 Introduction 9.2 Elastic behaviour of solids 9.3 Stress and strain 9.4 Hooke’s law 9.5 Stress-strain curve 9.6 Elastic moduli 9.7 Applications of elastic behaviour of materials
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Class 11 Physics Chapter 11 Thermal Properties of matter
Section Name Topic Name 11 Thermal Properties of matter 11.1 Introduction 11.2 Temperature and heat 11.3 Measurement of temperature 11.4 Ideal-gas equation and absolute temperature 11.5 Thermal expansion 11.6 Specific heat capacity 11.7 Calorimetry 11.8 Change of state 11.9 Heat transfer 11.10 Newton’s law of cooling
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Class 11 Physics Chapter 14 Oscillations
Section Name Topic Name 14 Oscillations 14.1 Introduction 14.2 Periodic and oscilatory 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
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Class 11th Physics Online Class For 100% Result
About Lesson

Introduction

In this chapter we will learn about oscillatory motion or oscillations. Any motion which repeats itself at regular intervals of time is known as periodic motion. If a body moves back and forth repeatedly about its mean position then it is said to be in oscillatory motion.

For example: The to and fro movement of pendulum, jumping on a trampoline, a child swinging on a swing.

Oscillations can be defined as Periodic to and fro motion which repeat itself at regular intervals of time.

Class 11 Physics Chapter 14 Oscillations Notes and NCERT Solution. www.free-education.in provide study material to excel in exam.
Oscillations Notes

Periodic and Oscillatory motions

Oscillations are defined as to and fro motion which repeat itself after regular intervals of time.In oscillations, the frequency of vibrations iscomparatively less. 

For example: The to and fro motion of a pendulum clock

Class 11 Physics Chapter 14 Oscillations Notes and NCERT Solution. www.free-education.in provide study material to excel in exam.
Class 11 Physics Chapter 14 Oscillations Notes and NCERT Solution. www.free-education.in provide study material to excel in exam.

Frequency

It is defined as number of cycles per second.

  • It is denoted by ν.
  • I.unit is sec-1
  • Special Unit is Hertz(Hz)
Class 11 Physics Chapter 14 Oscillations Notes and NCERT Solution. www.free-education.in provide study material to excel in exam.

Problem: – On an average human heart is found to be beat 75 times in a minute. Calculate frequency and time period?

Answer: –

The beat frequency of heart = 75/ (1 min)

 = 75/ (60 s)

 = 1.25 s–1

 = 1.25 Hz

 The time period T = 1/ (1.25 s–1) = 0.8 s 

Oscillations Notes

In the above image we can see that there is a block whose one end is attached to a spring and another is attached to a rigid wall.x is the displacement from the wall.

In the above figure a blockis attached to a spring, the other end of which is fixed to a rigid wall. The block moves on a frictionless surface. The motion of the block can be described in terms of its distance or displacement x from the wall.

f (t) = A cos ωt

As cosine function repeats after 2π so it can be written as

cos (θ) = cos (ωt + 2π)            Equation (1)

cos (ωt) = cos (ωt + 2π) (it keep on repeating after 2π)

Let Time Period = T

f (T) = f(t+T) where displacement keeps on repeating after (t+T)

Acos (ωt) = cosω(t+T) = Acos (ωt+ wT)

Acosωt = A cos (ωt+ωT)          Equation (2)

From Equation (1) and Equation (2)

ωT= 2π

Or T=2π /ω

Displacement as a combination of sine and cosine functions

f (t) = A cos ωt

f (t) = A sin ωt

f (t) = A sin ωt + A cos ωt

LetA = D cosΦ              Equation (3)

B=DsinΦEquation (4)

f (t) =DcosΦ sinωT + DsinΦ cos ωt

D (cosΦ sinωT + sinΦ cos ωt)

(Using sinAcosB + sinBcosA = sin (A+B))

Therefore we can write

f (T)= D sin (ωT+Φ)

From the above expression we can say displacement can be written as sine and cosine functions.

D in terms of A and B:-

AB= D2sin2 Φ + D2cos2 Φ

AB= D2

Or D= AB

Φ In terms of A and B

Dividing Equation (4) by (3)

B/A= DsinΦ/Dcos Φ

tan Φ = B/A

Or Φ= tan-1 B/A

Problem:-Which of the followingfunctions of time represent (a) periodic and (b) non-periodic motion? Give the period foreach case of periodic motion [ω is anypositive constant].

(i) sin ωt + cos ωt

(ii) sin ωt + cos 2 ωt + sin 4 ωt

(iii) e–ωt

(iv) log (ωt)

 Answer:-

  • sin ωt + cos ωt is a periodic function, it can also be written as

 2 sin (ωt + π/4).

Now 2 sin (ωt + π/4)= 2 sin (ωt + π/4+2π)

= 2 sin [ω (t + 2π/ω) + π/4]

The periodic time of the function is 2π/ω.

(ii) This is an example of a periodic motion. Itcan be noted that each term represents aperiodic function with a different angularfrequency. Since period is the least intervalof time after which a function repeats itsvalue, sin ωt has a period T0= 2π/ω; cos 2 ωt

has a period π/ω =T0/2; and sin 4 ωt has aperiod 2π/4ω = T0/4. The period of the firstterm is a multiple of the periods of the last

two terms. Therefore, the smallestintervalof time after which the sum of the threeterms repeats is T0, and thus the sum is aperiodic function with aperiod 2π/ω.

(iii) The function e–ωt is not periodic, itdecreases monotonically with increasingtime and tends to zero as t →∞ and thus,never repeats its value.

(iv) The function log (ωt) increases monotonicallywith time t. It, therefore, neverrepeats its value and is a non-periodicfunction. It may be noted that as t →∞,log (ωt) diverges to ∞. It, therefore, cannot represent any kind of displacement.

Class 11 Physics Chapter 14 Oscillations Notes and NCERT Solution. www.free-education.in provide study material to excel in exam.
Class 11 Physics Chapter 14 Oscillations Notes and NCERT Solution. www.free-education.in provide study material to excel in exam.

Phase

It is that quantity that determines the state of motion of the particle.

  • Its value is (ωt + Φ)
  • It is dependent on time.

Value of phase at time t=0, is termed as Phase Constant. When the motion of the particle starts it goes to one of the extreme position at that time phase is considered as 0.

Let x (t) = A cos (ωt) where we are taking (Φ = 0)

  1. Mean Position (t= 0)
  2. x (0) = A cos (0) = A (cos0=1)
  3. t=T/4, t= T/2, t=3T/4, t=T and t=5T/4
Class 11 Physics Chapter 14 Oscillations Notes and NCERT Solution. www.free-education.in provide study material to excel in exam.
Class 11 Physics Chapter 14 Oscillations Notes and NCERT Solution. www.free-education.in provide study material to excel in exam.
Oscillations Notes

Answer:

In SHM, acceleration a is related to displacement by the relation of the form

 a = -kx, which is for relation (c).

Problem: – The motion of a particle executing simple harmonic motion is described by the displacement function,

x (t) = A cos (ωt + 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 (ωt + α), what are the amplitude and initial phase of the particle with the above initial conditions.

Answer:-

Initially, at t = 0;

Displacement, x = 1 cm

Initial velocity, v = ω cm/ sec.

Angular frequency, ω = π rad/s–1

It is given that,

x (t) = A cos (ωt + Φ)

1 = A cos (ω × 0 + Φ) = A cos Φ

A cosΦ = 1                          … (i)

Velocity, v= dx/dt

ω = -A ω sin (ωt + Φ)

1 = -A sin (ω × 0 + Φ) = -A sin Φ

A sin Φ = -1                         … (ii)

Squaring and adding equations (i) and (ii), we get:

A2 (sin2 Φ + cos2 Φ) = 1 + 1

A2 = 2

∴ A = √2 cm

Dividing equation (ii) by equation (i), we get:

tan Φ = -1

∴Φ = 3π/4, 7π/4…

SHM is given as:

x = B sin (ωt + α)

Putting the given values in this equation, we get:

1 = B sin [ω × 0 + α] = 1 + 1

B sin α = 1                           … (iii)

Velocity, v = ω B cos (ωt + α)

Substituting the given values, we get:

π = π B sin α

B sin α = 1                            … (iv)

Squaring and adding equations (iii) and (iv), we get:

B2 [sin2 α + cos2 α] = 1 + 1

B2 = 2

∴ B = √2 cm

Dividing equation (iii) by equation (iv), we get:

B sin α / B cos α = 1/1

tan α = 1 = tan π/4

∴α = π/4, 5π/4…

Angular Frequency (ω)

Angular frequency refers to the angular displacement per unit time. It can also be defined as the rate of change of the phase of a sinusoidal waveform (e.g., in oscillations and waves). Angular frequency is larger than frequency ν (in cycles per second, also called Hz), by a factor of 2π.

Consider the oscillatory motion which is varying with time t and displacement x of the particle from the origin:

x (t) = cos (ωt + Φ)

Let Φ = 0

x (t) = cos (ωt)

After t=T i.e. x (t) = x (t+T)

A cos ωt = A cos ω (t + T)

Now the cosine function is periodic with period 2π, i.e., it first repeats itself after 2π. Therefore,

ω (t + T) = ωt + 2π

i.e. ω = 2π/ T

Where ω = angular frequency of SHM.

  • I. unit is radians per second.
  • It is 2π times the frequency of oscillation.
  • Two simple harmonic motions may have the same A and φ, but different ω.

Problem: – 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 ωt – cos ωt

(b) sin3ωt

(c) 3 cos (π/4 – 2ωt)

(d) cos ωt + cos 3ωt + cos 5ωt

(e) exp (–ω2t2)

Answer: –

(a) SHM

The given function is:

sinωt – cos ωt

This function represents SHM as it can be written in the form: a sin (ωt + Φ)

Its period is: 2π/ω

(b) Periodic but not SHM

The given function is:

sin 3ωt = 1/4 [3 sin ωt – sin3ωt]

The terms sin ωt and sin ωt individually represent simple harmonic motion (SHM). However, the superposition of two SHM is periodic and not simple harmonic.

Its period is: 2π/ω

(c) SHM

The given function is:

This function represents simple harmonic motion because it can be written in the form: a cos (ωt + Φ) its period is: 2π/2ω = π/ω

(d) Periodic, but not SHM

The given function is cosωt + cos3ωt + cos5ωt. Each individual cosine function represents SHM. However, the superposition of three simple harmonic motions is periodic, but not simple harmonic. 

(e) Non-periodic motion

The given function exp (-ω2t2) is an exponential function. Exponential functions do not repeat themselves. Therefore, it is a non-periodic motion.

 (f) The given function 1 + ωt + ω2t2 is non-periodic.

ProblemWhich of the followingfunctions of time represent (a) simple harmonic motion and (b) periodic but notsimple harmonic? Give the period for each case?

(1) sin ωt – cos ωt

(2) sin2 ωt

Answer:

(a) sin ωt – cos ωt

= sin ωt – sin (π/2 – ωt)

= 2 cos (π/4) sin (ωt – π/4)

 = √2 sin (ωt – π/4)

This function represents a simple harmonicmotion having a period T = 2π/ω and aphase angle (–π/4) or (7π/4).

(b) sin2 ωt

= ½ – ½ cos 2 ωt

The function is periodic having a period T = π/ω. It also represents a harmonic motion with the point of equilibrium occurring at ½ instead of zero.

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