**Class 11 Physics Chapter 14 Oscillations**

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**NCERT Notes For Class 11 Physics Chapter 14 Oscillations**

**Class 11 Physics Chapter 14 Oscillations**

**OSCILLATIONS**

## PERIODIC MOTION

- A motion that repeats at regular intervals of time.

Examples:

- Spinning of earth about its own axis
- Revolution of earth around the sun
- Oscillations of a pendulum
- Vibrations of a tuning fork

## Types of periodic motion

**Rotatorymotion:-** particle completes the rotation in regular intervals.

- Examples :-
- Rotation of earth around the sun
- Rotatory motion of hour hand, minute hand etc.

**Oscillatory motion** :- the particle moves to and fro with less frequency.

• Examples :- Bob of a pendulum

• Movement of swing

**Vibratory motion:-** the particle moves to and fro with large frequency.

**• Examples :-**

- The particle on a vibrating string
- Vibrations of atoms in a solid

### Period (T)

- The time taken to repeat a periodic motion is called the period (T).
- Its SI unit is second.
- The period of vibrations of a quartz crystal is expressed in units of microseconds (10
^{-6}s) abbreviated as μs. - The orbital period of the planet Mercury is 88 earth days.
- The Halley’s Comet appears after every 76 years.

### Frequency (ν)

- The number of repetitions in one second of a periodic motion is called Frequency (ν).
- Its unit is Hertz (Hz).
- The relation between v and T is, v

**Relation connecting period (T) , angular velocity (ω) and Frequency (ν)**

- The angular velocity, ω= 2πν
- Or ν=
- The period T = (1/ ν) = 2 π/ ω

## SIMPLE HARMONIC MOTION (SHM)

- In SHM the restoring force on the oscillating body is directly proportional to its displacement from the mean position and is directed opposite to the displacement.
- Eg: small oscillations of simple pendulum, swing, loaded spring, etc.

### DISPLACEMENT OF SHM

- The displacement is given by

*x*(*t* ) = *A*cos(ω*t* +φ)

### Displacement – Time graph of SHM

**Amplitude(A):**

- It is the magnitude of the maximum displacement of the oscillating particle Phase:
- The time varying quantity, (ωt + φ ), is called the phase of the motion.
- Phase describes the state of motion at a given time.

**Phase constant (or phase angle):**

- The constant φ is called the phase constant.
- The value of φ depends on the displacement and velocity of the particle at t = 0.

**Angular frequency (ω):**

- The SI unit of angular frequency is radians per second.

## SIMPLE HARMONIC MOTION AND UNIFORM CIRCULAR MOTION

• Simple harmonic motion is the projection of uniform circular motion on a diameter of the circle.

## VELOCITY IN SIMPLE HARMONIC MOTION

- The displacement of SHM is given by

- Differentiating with respect to time , we get the velocity

## Case I

- At mean position, x=0
- Then magnitude of velocity

*v*_{max }=ω*A*

**Case II**

- At extreme positions,
*x*=±*A* - Then
*v*_{min }=0

### ACCELERATION IN SHM

- A particle executing a uniform circular motion is subjected to a radial acceleration a directed towards the centre.
- Differentiating the velocity we get the acceleration as

- Thus in SHM, the acceleration is proportional to the displacement and is always directed towards the mean position.

**Case I**

- At mean position , x=0
- Thus, acceleration
*a*^{m in }=*o*

**Case II**

• **At the extreme positions , x=±A**

*a *max = − *A*ω^{2 }

**Graphs**

## FORCE LAW FOR SIMPLE HARMONIC MOTION

- The force law of SHM is
*F*(*t*) =−*kx*(*t*) , - where
*k*=*m*ω^{2 }is the force constant.

**Derivation**

### Linear harmonic oscillator

• The oscillator for which restoring force is a linear function of x.

### Non-linear harmonic or anharmonic oscillators

• Oscillators in which the restoring force is a nonlinear function of x .

### DIFFERENTIAL EQUATION OF SHM

- The restoring force acting on a particle in SHM is given by , F = -kx
- From Newton’s law we have F=ma

- That is

- For an angular displacement θ, the differential equation is,

## ENERGY IN SIMPLE HARMONIC MOTION

• A particle executing simple harmonic motion has kinetic and potential energies

### Kinetic Energy (K)

• The kinetic energy (K) is

### Derivation

- We have, the kinetic energy
*K*=*mv*^{2} - Substituting for velocity we get

- Thus kinetic energy is a periodic function of time.
- Kinetic energy is zero when the displacement is maximum and maximum at the mean position.
- The period of kinetic energy is T/2.

### Potential Energy(PE)

• The potential energy of SHM is

### Derivation

- The potential energy is given by
*U*=*kx*^{2}

- The potential energy of a particle executing simple harmonic motion is periodic, with period T/2.
- The potential energy is zero at the mean position and maximum at the extreme displacements.

### Total Energy(E)

- The total energy, E, of SHM is ,
*E*=*kA*^{2}

Derivation

- We have the the total energy
*E*= +*K U* - Thus

• The total mechanical energy of a harmonic oscillator is a constant.

### Energy –Time graph of SHM

### Energy –Displacement graph of SHM

### EXAMPLES OF SIMPLE HARMONIC MOTION Oscillations due to a Spring

- If the block is pulled on one side and is released, it then executes a to and fro motion about a mean position.
- At any time t, the restoring force F acting on the block is,

- The constant of proportionality, k, is called the spring constant.
- A stiff spring has large k and a soft spring has small k.
- The equation is same as the force law for SHM and therefore the system executes a simple harmonic motion.
- The frequency of oscillations is ω=√
*k/**m*

- The period is

## The Simple Pendulum

- A simple pendulum, consists of a particle of mass m (called the bob of the pendulum) suspended from one end of an unstretchable, massless string of length L fixed at the other end.

**The forces acting on the bob are:**

- Tension in the string
- Gravitational force.

### Expression for time period

- The string makes an angle θ with the vertical.
- We resolve the force F
_{g }into a radial component F_{g }cos θ and a tangential component Fg sin θ. - The radial component of force F
_{g }cos θ, is cancelled by the tension. - The tangential component, F
_{g}sin θ produces a restoring torque . - The restoring torque τ is τ=−
*LF*sinθ_{g } - Where the negative sign indicates that the torque acts to reduce θ.
- For rotational motion we haveτ α=
*I* - where I is the pendulum’s moment of inertia about the pivot point and α is its angular acceleration about that point.

- That is, the angular acceleration of the pendulum is proportional to the angular displacement θ but opposite in sign.
- Thus the motion of a simple pendulum swinging through small angles is approximately SHM.

The angular frequency

### DAMPED SIMPLE HARMONIC MOTION

- The SHM which dies out due to the dissipative forces acting on it is called damped simple harmonic oscillation.
- Eg: free oscillations of a simple pendulum.

### Differential equation of Damped oscillations

- Damped SHM is given by the equation

- Where m- mass, b –damping constant, k- spring constant

#### Derivation

• The damping force (viscous force) exerted by the liquid on the system is

*F _{d }*=−

*bv*

- where b is a damping constant .
- The negative sign shows that the force is opposite to the velocity.
- The restoring force on the spring is

*F _{S }*=−

*kx*

- Thus the total force acting on the mass at any time t is

*F *=−*kx* −* bv*

- If a is the acceleration of the mass at time t, then by Newton’s second law of motion

### Solution of the differential equation

- The solution of this equation is of the form

- Where

### Amplitude of damped oscillator

- The amplitude is given by

- Thus amplitude decreases with time.

### Undamped oscillator

• If b=0(there is no damping), then *x t*( ) = *A*cos(ω φ′*t *+ )

### Energy of a damped oscillator

- If the oscillator is damped, the mechanical energy decreases with time.
- The total energy is given by

### Displacement – time graph of damped oscillator

## Free Oscillations

- The oscillations of a body oscillated and left free, are called free oscillations.
- The frequency of free oscillation is known as natural frequency.
- A person swinging in a swing without anyone pushing it or a simple pendulum, displaced and released, are examples of free oscillations.

### FORCED OSCILLATIONS

- When a periodic force is applied to maintain an oscillation, it is known as forced or driven oscillations.
- The frequency of forced oscillations is equal to the frequency of the applied force.
- The two angular frequencies in a forced oscillation are
- The natural angular frequency ω of the system
- The angular frequency ω
_{d}of the external force causing the driven oscillations. - The oscillator initially oscillates with its natural frequency ω.
- When we apply the external periodic force, the oscillations with the natural frequency die out, and then the body oscillates with the (angular) frequency of the external periodic force.

### Differential equation of forced oscillation

- The forced oscillation is given by

- Where F
_{0}– amplitude of periodic force. - The periodic force is

### Derivation

• The motion of a particle under the combined action of a linear restoring force, damping force and a time dependent driving force is given by

• Or

### Solution of the Differential equation

The solution of the equation is

• Where

- And

- Where m – mass of the particle, v
_{0}and x_{0}are the velocity and the displacement of the particle at time t = 0.

### Special Cases

1. Small Damping, Driving Frequency far from Natural Frequency :

- In this case, ω
_{d}b will be much smaller than m(ω^{2}–ω^{2}d) and we can neglect that term.

- If we go on changing the driving frequency, the amplitude tends to infinity when it equals the natural frequency

**2. Driving Frequency Close to Natural Frequency**

- If ω
_{d}is very close to ω ,then m(ω^{2}–ω^{2}d) would be much less than ω_{d}b. - Therefore

- Thus the maximum possible amplitude for a given driving frequency is governed by the driving frequency and the damping.

### Resonance

- The phenomenon of increase in amplitude when the driving force is close to the natural frequency of the oscillator is called resonance.
- The Tacoma Narrows Bridge, USA collapsed due to resonance..
- To avoid rupture of bridges due to resonance , marching soldiers break steps while crossing a bridge.
- Aircraft designers make sure that none of the natural frequencies at which a wing can oscillate match the frequency of the engines in flight.
- In an earthquake, short and tall structures remain unaffected while the medium height structures fall down.

- The natural frequencies of the short structures are higher and those of taller structures lower than the frequency of the seismic waves.