Chapter – 18, Wave Motion

Introduction : If we through a stone in a quiet pond, we can observe the following shape.

It may be considered as wave. If one person say something then the sound disturbs the layers of the air between the two persons and it may be assumed like a scene after drawing of a book from the bookshelf. We have already understood what wave is in class IX. So, in present chapter, we will understand the wave motion and its application to our daily life.

Wave Motion :  The propagation of disturbance is called as wave motion.

Some important terms related to Wave motion :

1)  Crest : It is the point on the wave at the  maximum upward displacement with in a cycle  from the mean position.

2) Trough : It is the point on the wave at the maximum downward displacement with in a cycle from the mean position.

3 ) Amplitude : It is the maximum displacement of the particle from the mean position. It is the distance between the mean position and a crest or a trough of a wave. Its units are m andcm.

4 ) Wavelength : It is the distance between the two consecutive crests or two consecutive trough of a wave. Its units are m and cm.

5 ) Phase : The angular representation of a particle in a wave is called the phase. It is the property of a wave.

6 ) Epoch : It is initial representation of a particle in a wave.

7 ) Path : It is the linear path traveled by a particle in wave motion.

Note : The phase of a vibrating particle changes continuously with time, but the epoch of the particle is independent of time and it remains constant.

8 ) Phase Difference : The difference between the phases of a particle at two different positions in the wave motion is termed as the phase difference.

9 ) Path Difference : The difference between the paths of a particle at two different positions in wave motion is termed as the Path difference.

10 ) Relation between the path difference and the phase difference :

When the path difference is λ, the phase difference = 2π

when the path difference is 1, the phase difference = 2π / λ

when the path difference is x, the phase difference = 2πx / λ

Hence, Ф =  2πx / λ

Periodic Motion : The motion which repeats itself at regular intervals of time, is called as Periodic Motion.

Oscillation : If a particle that executes periodic motion moves to and fro repeatedly about a mean position, the motion is called as Oscillation.

An Oscillation is always a periodic motion but a periodic motion may or may not be oscillation.

Harmonic Oscillation : The oscillation which can be expressed in terms of single harmonic function, is termed as Harmonic Oscillation.

Simple Harmonic Motion 

A harmonic oscillation of constant amplitude and single frequency is called as Simple Harmonic Motion.

It is defined as the oscillation about a fixed point ( mean position ) in which restoring force is always proportional to the displacement from that point and always directed from away from that point.

Any function which repeats itself after regular intervals of time is called periodic function.

Examples : f ( t ) = sinωt and g ( t ) = cosωt

where ω = 2π / T

and T is the time period of oscillation. 

Description of Simple Harmonic Motion :

1 ) Equations of Simple Harmonic Motion or equation of displacement :

a ) x = a sinωt 

b ) x = a sin( ωt – ϕ )

c ) x = a sin( ωt + ϕ )

d ) x = a cosωt

e ) x = a cos( ωt – ϕ )

f ) x = a cos( ωt +ϕ )

where ϕ is epoch of the particle and ω = 2π / T, T is the time period of the simple harmonic motion. 

Note : 

i ) If time and parameter of simple harmonic motion are counted from mean position then sine function is used 

ii ) If the time and parameter are counted from extreme position, cosine function is used.

2 ) Velocity : 

Let the displacement of the particle executing Simple harmonic motion having angular frequency ω and time period T is given by

x = a cosωt ——————————— ( 1 ) 

sinωt = √ ( 1 – cos2ωt ) = √ ( 1 – x2 / a2 ) = { √ ( a2 – x2 ) } / a 

differentiating both sides of equation ( 1 ) with respect to time ‘ t ‘ , we get

dx / dt = a ( – sinωt )ω

v = – aωsinωt

v = –ω√ ( a2 – x2 )

Here, negative sign indicates that the direction of the velocity is opposite of displacement.

Note :

i ) When x = a, then

v = –ω√ ( a2 – a2 ) = 0 

That is the velocity of the particle is zero when it is at extreme position.

ii ) when x = 0, then,

v = –ω√ ( a2 – 02 )

v = – aω

The particle has maximum velocity at the mean position and its direction is opposite of its displacement.

3 ) Acceleration : 

Let the displacement of the particle executing Simple harmonic motion having angular frequency ω and time period T is given by

x = a cosωt ——————————— ( 1 ) 

sinωt = √ ( 1 – cos2ωt ) = √ ( 1 – x2 / a2 ) = { √ ( a2 – x2 ) } / a 

differentiating both sides of equation ( 1 ) with respect to time ‘ t ‘ , we get

dx / dt = a ( – sinωt )ω

v = – aωsinωt

Again differentiating with respect to ‘ t ‘ we get

dv / dt = -aω ( cosωt ) ω

f = -aω2cosωt

Using equation ( 1 )

f = -aω2x

Note :

i ) when x = a ( extreme position )

f = -a2ω2

Hence, the particle has maximum acceleration at the extreme position and directed away from the mean position.

ii ) when x = 0 ( mean position )

f = 0

Hence, the particle has minimum acceleration at mean position.

4 ) Differential form of simple harmonic motion : 

Let the displacement of the particle executing Simple harmonic motion having angular frequency ω and time period T is given by

x = a cosωt ——————————— ( 1 ) 

differentiating both sides of equation ( 1 ) with respect to time ‘ t ‘ , we get

dx / dt = a ( – sinωt )ω

dx / dt = – aωsinωt

Again differentiating with respect to ‘ t ‘ we get

d2x / dt2 = -aω ( cosωt ) ω

d2x / dt2 = -aω2x

d2x / dt2 + aω2x = 0

5 ) Restoring Force : 

Restoring force brings the particle back to its initial position and it is given by

P = mass * acceleration

P = m * f

P = m ( – a2ω2 x )

P = – ma2ω2 x

6 ) Energy : 

Let x = displacement of the particle

v = velocity of the particle = –ω√ ( a2 – x2 )

a = amplitude

ω = angular frequency = 2π / T 

T = time period 

P = restoring force =  ma2ω2 

Potential Energy ( EP ) = Average force * displacement

EP = { ( 0 + P ) / 2 } * x

EP = ma2ω2 x2 / 2

Kinetic energy ( Ek ) = mv2 / 2

Ek = m { –ω√ ( a2 – x2 ) }2 / 2

Ek = m ω2 ( a2 – x2 ) / 2

The total energy is given by

E = EP + Ek

E = ma2ω2 x2 / 2 +  m ω2 ( a2 – x2 ) / 2

E = ma2ω2 / 2  = constant

7 ) Characteristics Of Simple Harmonic Motion : 

i ) It is the harmonic motion of constant amplitude and single frequency.

ii ) It is the projection of uniform circular motion on a diameter of the circle of reference.

iii ) A particle in simple harmonic motion has maximum velocity at mean position and zero velocity at extreme position.

iv ) A particle in simple harmonic motion has zero acceleration at mean position and maximum acceleration at extreme position.

v ) Simple harmonic motion executes maximum acceleration having zero velocity and vice – versa.

vi ) It is to and fro motion in nature.

vii ) The acceleration of a particle executing simple harmonic motion is directly proportional to its displacement from the mean position.

viii ) The acceleration of a particle executing simple harmonic motion is directed towards the mean position.

ix ) The total energy of a body executing simple harmonic motion is constant.

x ) Mass of the particle and force constant affect the time period and the frequency of the simple harmonic motion.

                                                                                                

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