CBSE Class 11 Physics Revision Notes Chapter 14

Class 11 Physics Revision Notes for Chapter 14 – Oscillations 

The revision notes of Chapter 14- Oscillations provided by Extramarks will help students get a clear idea of all the important topics covered in this chapter. The subject-matter experts have curated Class 11 Physics Chapter 14 Notes as per the latest syllabus of the CBSE for students who are preparing for their Class 11 exam.

Class 11 Physics Revision Notes for Chapter 14 – Oscillations – Free Download

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Access Class 11 Physics Chapter 14 – Oscillation and Waves 

INTRODUCTION:

• Periodic motion is the type of motion that repeats itself after regular intervals of time over and over again. 
• The oscillatory or vibratory motion is the type of motion in which an object moves to and fro or back and forth in a definite interval of time in a repetitive manner about a fixed point.
• A specific kind of oscillatory motion is simple harmonic motion, which includes the following:

1. the particle moves are restricted to a single dimension
2. Whenever Fnet=0, the particle oscillates back and forth around a constant mean location.
3. The equilibrium position is always the direction of the net force acting on the particle.
4. The particle’s displacement from the equilibrium location at that instant is always inversely proportional to the size of the net force.

1.1 Some Important Terms:

• The amplitude of a particle executing S.H.M. is its maximum displacement on either side of the equilibrium position. It is represented by A.
• The time period of a particle executing S.H.M. is the time taken to complete one cycle and is represented by T.
• The frequency of a particle executing S.H.M. is denoted by v and is the same as the number of oscillations completed in one second.
• The phase of particles executing S.H.M. at any instant is its state with respect to the direction of motion and its position at that particular instant. Phase=(ωt+ϕ)

1.2 Important Relations:

CBSE Class 11 Physics Chapter 14 Notes explain the following in detail: 

1. Position– When the equilibrium position is at origin, the position depends on time in general as x (t)=sin (ωt+ϕ)

At the equilibrium position, x = 0 while at the extremes, x=+a,−a

2. Velocity- 

• v(t)=Aωcos(ωt +ϕ) at any instant t.
• v(x)=±ω√(A2−x2) at any position x.
• Since the particle is at rest at the extremes, Velocity has minimum magnitude here i.e. v=0 at the extreme position. 
• Velocity has maximum magnitude at the equilibrium position i.e.|v|max=ωA here. 

3. Acceleration-

• a(t)=ω2A sin(ωt+ϕ) at any instant t.
• a(x)=ω2x at any position x.
• It is always directed towards the equilibrium position. 
• The magnitude of the acceleration is minimum at the equilibrium position i.e. |a|min=0 and maximum at extremes i.e. |a|max=ω2A 

Potential Energy:

• If the potential energy is considered as zero at the equilibrium position, then at any position x, U(x)=½ kx2=½ mA2ω2sin2(ωt+ϕ) 
• U is maximum at extremes, Umax=½ kA2
• Potential energy is minimum at the equilibrium position.

Total Energy:

T.E.=½ kA2=½ mA2ω2 and it is constant at all positions and instants of time.

Energy Position Graph:

Kinetic energy, potential energy and total energy are all functions of time in an energy position graph. Both potential energy and kinetic energy repeats after time T/2. 

2. TIME PERIOD OF S.H.M:

For understanding whether a motion is S.H.M. or not and to compute its time period, the following are the steps: 

1. Mathematically locate the equilibrium position by balancing all the forces acting on it. 
2. Move the particle by a displacement ‘x’ from its mean position in the most likely direction of oscillation. 
3. Analyse the net force acting on it to see if it is moving toward the mean position.
4. Try to express net force as a proportional function of its displacement ‘x’. 
   • If steps (c) and (d) are established, the motion is a basic harmonic motion.
5. Determine k from the expression of net force (F=kx) and find the time period using T=2π√(m/k)

2.1 Oscillations of a Block Connected to a Spring:

a) Horizontal spring:

Consider placing a block of mass m on a flat, smooth surface and rigidly attaching it to the other end of a spring with a force constant K that is fixed in place.

• Mean position: the spring in this position is at its natural length.
• Time period: T=2π√(m/k)

b) Vertical Spring:

If the spring is suspended vertically from a fixed point while being carried by its opposite end as shown, the block will oscillate along the vertical line.

• Mean position: Via d=(mg)/k, the spring in elongated
• Time period: T=2π√(m/k)

c) Combination of springs: 

• Springs in series: 

Take into account two springs with the corresponding force constants K1 and K2, connected in series as indicated. They are comparable to one spring with force constant K, denoted by

1/K=1/K1+1/K2  

⇒K=K1K2/K1+K2 

• Springs in parallel:

The effective spring constant for a parallel combination is K=K1+ K2

2.2 Oscillation of a Cylinder Floating in a Liquid:

Suppose the total length of the cylinder is L and a cylinder of mass m and density d be floating on the surface of a liquid of density ρ.

• Mean position: It is where the cylinder is immersed up to ℓ=(Ld)/ρ
• Time period: T=2π√[(Ld)(ρg)] =2π√(ℓ/g) 

2.3 Liquid Oscillating in a U–Tube:

In a U-tube of the area of cross-section A, suppose a liquid column of mass m and density ρ.

Mean position: It is when the height of the liquid is the same in both limbs.

Time period:  T=2π√[m/(2Aρg)]= 2π√L/(2g)

2.4 Body Oscillation in the Tunnel Along any Chord of the Earth:

• Mean position: It is at the centre of the chord 
• Time period: T=2π√(Rg) = 84.6 minutes where the radius of the earth is R.

2.5 Angular Oscillations:

When a centre or particle of mass of a body oscillates on a small arc of the circular path, instead of straight-line motion, then it is called an angular S.H.M.  

For angular S.H.M., torque is:

τ=kθ

where θ is the angular displacement and k is a constant.

⇒ Iα=−kθ

where α is the angular acceleration and I is the moment of inertia.

⇒T=2π√(I/k), is the time period of oscillations

2.5.1 Simple Pendulum:

• Time period: T=2π√(ℓ/g)  
• Time period of a pendulum in a lift: 

If the acceleration of lift acts upwards T=2π√[ℓ/(g+a)]

If the acceleration of lift acts downwards T=2π√[ℓ/(g-a)]

• For a second’s pendulum: 

2s is the time period of the second’s pendulum. 

≃1m is the length of a second’s pendulum on the earth’s surface.

2.5.2 Physical Pendulum:

Time period: T=2π√[I/(mgℓ)]

I denotes the body’s moment of inertia with respect to the point of suspension, and ℓ denotes the distance between the body’s centre of mass and the point of suspension.

3. DAMPED AND FORCED OSCILLATIONS:

• A body can vibrate in a damped oscillation when the amplitude of the vibration gradually decreases over time.
• In this kind of vibration, damping forces including frictional force and viscous force, among others, cause the amplitude to drop exponentially.
• As the amplitude declines, the oscillator’s energy also continues to drop off exponentially.
• Forced oscillation is a type of vibration in which a body vibrates in response to a periodic external force.
• Resonance: When the frequency of an external force and the natural frequency of an oscillator match, this is referred to as the resonance condition. The term “resonant frequency” refers to this same frequency.

4. WAVES

Speed of a Longitudinal Wave: 

• Speed of a longitudinal wave is mathematically represented as: v=√(E/ρ) in a medium, where ρ is the density of the medium and E is the modulus of elasticity.
• Speed of a longitudinal wave is represented as v=√(Y/ρ) in a solid in the form of the rod where Y is Young’s modulus of the solid and ρ is the density of the solid.
• Speed of longitudinal waves is represented as v=√(B/ρ) in a fluid, where, ρ is the density of the fluid and B is the bulk modulus.

Newton’s Formula: 

• Newton stated that the propagation of the sound waves (in gas) is an isothermal process. The speed of sound in a gas is given by v=√(P/ρ) according to him, where ρ is the density of the gas and P is the pressure of the gas. 
• The speed of sound in air at S.T.P. is 280m/s according to Newton’s formula. However, 332ms−1 is the experimental value of the speed of sound in air. Newton could not explain this large difference in time and his formula was later rectified by Laplace. 

Laplace’s Correction: 

• Laplace claimed that sound waves (in gas) propagate through an adiabatic process. He claimed that the formula for the speed of sound in a gas is v=√[(γP)/ρ], where γ is the ratio of specific heats.  
• He claims that the speed of sound in air at the S.T.P. is 331.3 m/s, which agrees fairly well with the experimental data.

5. WAVES TRAVELLING IN OPPOSITE DIRECTIONS:

Think of two waves that are travelling in opposing directions with identical amplitude and frequency:

y1=A sin(kxωt)

y2=A sin(kx +ωt)

Considering them to be interfering, a standing wave is produced as a result of:

y=y1+y2

⇒y=2A sin kx cos ωt

It is obvious that the particle at position x is oscillating in S.H.M. with angular frequency and amplitude 2A sin kx. Due to the location-dependent nature of the amplitude, particles oscillate at different intensities (x).

• Nodes: 

Amplitude = 0

⇒2A sin kx=0 

⇒x=0,π/k,2π/k….  

⇒x=0,λ/2,λ,3λ/2,2λ….  

• Antinodes:  

Amplitude is maximum.

sin kx=±1

⇒x=π/(2k),3π/(2k)

⇒x=λ4,3λ4,5λ4

5.1 Vibrations in a Stretched String:

Fixed at Both Ends: 

• Transverse standing waves are formed with nodes at both ends of the string. 
• Length of string: ℓ=nλ/2,  if there are 

(n + 1) nodes and n antinodes. 

• Frequency of oscillations: ν=v/λ=nv/(2ℓ)
• Fundamental frequency (x=1) or first harmonic: ν0= v/(2L)
• Second Harmonic or First Overtone: ν=(2v)/(2L)=2ν0 
• The nth multiple of the fundamental frequency is known as the nth harmonic or (n1)th overtone.

Fixed at One End:

• Transverse standing waves with an antinode at the open end and node at the fixed end are formed. 
• Length of string: ℓ=(2n−1)λ/4, if there are n nodes and n antinodes. 
• Frequency of oscillations: ν=v/λ=[(2n−1)v]/(4ℓ)
• Fundamental frequency (n=1) or first harmonic: ν0=v/(4L)
• First overtone or third harmonic 

ν=(3v)/(4ℓ)=3ν0 

• In this case, only odd harmonics are possible.

5.2 Vibrations in an Organ Pipe:

Open Organ pipe (both ends open): 

• The open ends of the tube generate antinodes because the particles there are free to vibrate.
• When there are (n+1) antinodes in total, the length of tube: ℓ=(nλ)/2
• Frequency of oscillations: ν=(nv)/(2ℓ)

Closed Organ Pipe (One end closed): 

• The closed end forms a node and the open end forms an antinode. 
• When there are n antinodes and n nodes,  L=(2n−1)λ/(4L).
• Frequency of oscillations: ν=v/λ=(2n−1)v/(4L).

5.3 Waves Having Different Frequencies:

Beats are created by superimposing two waves that are propagating in the same direction but at a slightly different frequency. At any fixed location, the effect in this situation will consist of alternate loud and weak sounds.

6. DOPPLER EFFECT:

According to the Doppler effect, whenever there is a relative motion between a source of sound and a listener, the apparent frequency of the sound heard by the listener differs from the actual frequency of sound emitted by the source. The apparent frequency is mathematically given by: 

ν′=(ν−νL)/(v−νS) × ν

7. CHARACTERISTICS OF SOUND:

• Level of intensity of sound is another name for the loudness of sound. In decibels, a sound with intensity I is measured for loudness represented by  L=10log10(I/I0)

where, I0=10−12W/m2   

• The frequency of a sound affects its pitch. The louder and higher the pitch of a sound correspond to its higher frequency.

Class 11 Physics Revision Notes for Chapter 14 Oscillations 

The mechanism of parametric resonance is explained in Chapter 14 Physics Class 11 Notes. The damping of the oscillations is caused by some friction in reality. For oscillations to occur for parametric resonance, the amplification coefficient must exceed a certain minimum value consequently. Apart from this, these notes also provide a thorough understanding of all the other important concepts of the chapter. 

Examples 

The movement of a basic pendulum in a clock and the tides in the sea are two of the most typical oscillation examples. Another example is the motion of spring. Oscillations can also be seen in the vibrating of guitar strings and those of other stringed instruments. 

Damped Oscillation 

Damping is the technique of restricting or regulating oscillatory motion. When an oscillation is induced by a restoring force that is equal to the restraining force, it remains undamped and the system continues to oscillate with the same energy. However, a damped oscillation is not the best type of oscillation system. The amplitude of the oscillations does not alter with time, and neither does their magnitude. 

Refer to Notes of Class 11 Physics Chapter 14 for a detailed explanation. 

Simple Harmonic 

The most basic mechanical oscillating system consists of a weight connected to a linear spring that is under tension and weight. On an air table or an ice surface, one can approximate such a system. When the spring is static, the system is in a state of equilibrium.

The mass is moved back to the equilibrium point, but as it moves there, it gains momentum and continues to move past it, creating a fresh restoring force in the opposite sense. The displacement of the restoring force on a body is precisely proportional to it, just like the dynamics of the spring-mass system. A mass-spring system, which displays the existence of an equilibrium and the presence of a restoring force that becomes stronger the moment the system deviates from such equilibrium, demonstrates the common properties of oscillation.