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TYPES OF WAVE MOTION The mechanical waves are of two types.The mechanical waves are of two types.The mechanical waves are of two types.The mechanical waves are of two types. • Transverse wave motionTransverse wave motionTransverse wave motionTransverse wave motion • Longitudinal wave motionLongitudinal wave motionLongitudinal wave motionLongitudinal wave motion

Transverse wave motion- A transverse wave motion is that wave motion,A transverse wave motion is that wave motion,A transverse wave motion is that wave motion,A transverse wave motion is that wave motion, in which in which in which in which individual individual individual individual particles of the medium execute simple harmonic motion about their mean particles of the medium execute simple harmonic motion about their mean particles of the medium execute simple harmonic motion about their mean particles of the medium execute simple harmonic motion about their mean position in a direction perpendicular to the direction of propagation of position in a direction perpendicular to the direction of propagation of position in a direction perpendicular to the direction of propagation of position in a direction perpendicular to the direction of propagation of wave motion.wave motion.wave motion.wave motion.

For exampleFor exampleFor exampleFor example---- (i) (i) (i) (i) MovementMovementMovementMovement of string of a sitar or violinof string of a sitar or violinof string of a sitar or violinof string of a sitar or violin (ii(ii(ii(ii) Movement) Movement) Movement) Movement of membrane of a tablaof membrane of a tablaof membrane of a tablaof membrane of a tabla (iii(iii(iii(iii) Movement) Movement) Movement) Movement of a kink on a ropeof a kink on a ropeof a kink on a ropeof a kink on a rope

Waves set up on the surface of water are a combination of transverse Waves set up on the surface of water are a combination of transverse Waves set up on the surface of water are a combination of transverse Waves set up on the surface of water are a combination of transverse waves and longitudinal waves.waves and longitudinal waves.waves and longitudinal waves.waves and longitudinal waves. Light waves and all other electroLight waves and all other electroLight waves and all other electroLight waves and all other electro----magnetic magnetic magnetic magnetic waves are also transverse waves.waves are also transverse waves.waves are also transverse waves.waves are also transverse waves. A transverse wave travels through a A transverse wave travels through a A transverse wave travels through a A transverse wave travels through a medium in the form of medium in the form of medium in the form of medium in the form of crests and and and and troughs. A A A A crest is a portion of the medium which is raised temporarily above the is a portion of the medium which is raised temporarily above the is a portion of the medium which is raised temporarily above the is a portion of the medium which is raised temporarily above the normal position of rest of the particles of the medium,normal position of rest of the particles of the medium,normal position of rest of the particles of the medium,normal position of rest of the particles of the medium, when a transverse when a transverse when a transverse when a transverse wave passes through it.wave passes through it.wave passes through it.wave passes through it. The The The The ccccentreentreentreentre of crest is the position of maximum of crest is the position of maximum of crest is the position of maximum of crest is the position of maximum displacement in the positive direction.displacement in the positive direction.displacement in the positive direction.displacement in the positive direction. A A A A trough is a portion of the medium which is depressed temporarily is a portion of the medium which is depressed temporarily is a portion of the medium which is depressed temporarily is a portion of the medium which is depressed temporarily below the normal position of rest of the particles of the medium,below the normal position of rest of the particles of the medium,below the normal position of rest of the particles of the medium,below the normal position of rest of the particles of the medium, when a when a when a when a transverse wave passes through it.transverse wave passes through it.transverse wave passes through it.transverse wave passes through it. The centre of trough is the position of The centre of trough is the position of The centre of trough is the position of The centre of trough is the position of maximum displacement in the negative dirmaximum displacement in the negative dirmaximum displacement in the negative dirmaximum displacement in the negative direction.ection.ection.ection. The distance between two consecutive crests or two consecutive troughs The distance between two consecutive crests or two consecutive troughs The distance between two consecutive crests or two consecutive troughs The distance between two consecutive crests or two consecutive troughs is called is called is called is called wavelength of the wave.of the wave.of the wave.of the wave. It is represented by It is represented by It is represented by It is represented by λλλλ Thus AC = BD = Thus AC = BD = Thus AC = BD = Thus AC = BD = λλλλ For the propagation of mechanical waves,For the propagation of mechanical waves,For the propagation of mechanical waves,For the propagation of mechanical waves, the material medium must the material medium must the material medium must the material medium must possess the following characteristics:possess the following characteristics:possess the following characteristics:possess the following characteristics: (i)(i)(i)(i)Elasticity,,,, so that particles can return to their mean position,so that particles can return to their mean position,so that particles can return to their mean position,so that particles can return to their mean position, after after after after having been disturbed.having been disturbed.having been disturbed.having been disturbed. (ii)(ii)(ii)(ii)Inertia, so that particles can store energy and overshoot their mean so that particles can store energy and overshoot their mean so that particles can store energy and overshoot their mean so that particles can store energy and overshoot their mean position.position.position.position.

SOME TERMS CONNECTED WITH WAVE

MOTION • Wavelength- Wavelength of a wave is the length of one wave.Wavelength of a wave is the length of one wave.Wavelength of a wave is the length of one wave.Wavelength of a wave is the length of one wave. It It It It

is equal to the distance travelled by the wave during the is equal to the distance travelled by the wave during the is equal to the distance travelled by the wave during the is equal to the distance travelled by the wave during the time;time;time;time; any one any one any one any one particle of the medium completes one vibration about its mean particle of the medium completes one vibration about its mean particle of the medium completes one vibration about its mean particle of the medium completes one vibration about its mean position.position.position.position. We may also define wavelength as the distance between We may also define wavelength as the distance between We may also define wavelength as the distance between We may also define wavelength as the distance between any two nearest particles of the medium,any two nearest particles of the medium,any two nearest particles of the medium,any two nearest particles of the medium, vibrating in the samevibrating in the samevibrating in the samevibrating in the same phase.phase.phase.phase.

As stated already transverse wave motion,As stated already transverse wave motion,As stated already transverse wave motion,As stated already transverse wave motion, λ λ λ λ = distance between = distance between = distance between = distance between centerscenterscenterscenters of two consecutive crests or of two consecutive crests or of two consecutive crests or of two consecutive crests or distance between distance between distance between distance between centerscenterscenterscenters of two consecutive troughs.of two consecutive troughs.of two consecutive troughs.of two consecutive troughs. Also,Also,Also,Also, wavelength can be taken as the distance in which one crest and one wavelength can be taken as the distance in which one crest and one wavelength can be taken as the distance in which one crest and one wavelength can be taken as the distance in which one crest and one trough are contained.trough are contained.trough are contained.trough are contained. Similarly, in a longitudinal wave motion,Similarly, in a longitudinal wave motion,Similarly, in a longitudinal wave motion,Similarly, in a longitudinal wave motion, λ λ λ λ = distance between the = distance between the = distance between the = distance between the centerscenterscenterscenters of two consecutive of two consecutive of two consecutive of two consecutive compressions or distance between two consecutive rarefactions.compressions or distance between two consecutive rarefactions.compressions or distance between two consecutive rarefactions.compressions or distance between two consecutive rarefactions. Also,Also,Also,Also, wavelength can be taken as the distance in which one wavelength can be taken as the distance in which one wavelength can be taken as the distance in which one wavelength can be taken as the distance in which one compression and one rarefaction are contained.compression and one rarefaction are contained.compression and one rarefaction are contained.compression and one rarefaction are contained. • Frequency-Frequency of vibration of a particle is defined as the Frequency of vibration of a particle is defined as the Frequency of vibration of a particle is defined as the Frequency of vibration of a particle is defined as the

number of vibrations completed by particle in one second.number of vibrations completed by particle in one second.number of vibrations completed by particle in one second.number of vibrations completed by particle in one second. As one As one As one As one vibration is equivalent to one wavelength,vibration is equivalent to one wavelength,vibration is equivalent to one wavelength,vibration is equivalent to one wavelength, therefore,therefore,therefore,therefore, we may define we may define we may define we may define frequency of a wave as the number of complete wavelengths frequency of a wave as the number of complete wavelengths frequency of a wave as the number of complete wavelengths frequency of a wave as the number of complete wavelengths transversetransversetransversetransversed by the wave in one second.d by the wave in one second.d by the wave in one second.d by the wave in one second. It is represented by It is represented by It is represented by It is represented by υυυυ....

• Time period-Time period of vibration of a particle is defined as Time period of vibration of a particle is defined as Time period of vibration of a particle is defined as Time period of vibration of a particle is defined as

the time taken by the particle to complete one vibration about its the time taken by the particle to complete one vibration about its the time taken by the particle to complete one vibration about its the time taken by the particle to complete one vibration about its mean position.mean position.mean position.mean position. As one vibration is equivalent to one wavelength,As one vibration is equivalent to one wavelength,As one vibration is equivalent to one wavelength,As one vibration is equivalent to one wavelength, therefore,therefore,therefore,therefore, time period of a wave is equal to time taken by the wave to time period of a wave is equal to time taken by the wave to time period of a wave is equal to time taken by the wave to time period of a wave is equal to time taken by the wave to travel a distance equal to one wavelength.travel a distance equal to one wavelength.travel a distance equal to one wavelength.travel a distance equal to one wavelength. It is represented by It is represented by It is represented by It is represented by TTTT....

RELATION BETWEEN υ AND T By definition,By definition,By definition,By definition, Time for completing v vibrations = 1 secTime for completing v vibrations = 1 secTime for completing v vibrations = 1 secTime for completing v vibrations = 1 sec Time for completing 1 vibration = 1/Time for completing 1 vibration = 1/Time for completing 1 vibration = 1/Time for completing 1 vibration = 1/υυυυ secsecsecsec i.e. i.e. i.e. i.e.

T = 1/T = 1/T = 1/T = 1/υυυυ or or or or υυυυ = 1/T or = 1/T or = 1/T or = 1/T or υυυυT = 1T = 1T = 1T = 1 ………….………….………….…………. (1)(1)(1)(1)

RELATION BETWEEN VELOCITY,

FREQUENCY AND WAVELENGTH OF A WAVE

Suppose Suppose Suppose Suppose υυυυ = frequency of a wave= frequency of a wave= frequency of a wave= frequency of a wave T = time period of the waveT = time period of the waveT = time period of the waveT = time period of the wave λ λ λ λ = wavelength of the wave= wavelength of the wave= wavelength of the wave= wavelength of the wave v = velocity of the wave.v = velocity of the wave.v = velocity of the wave.v = velocity of the wave. By definitionBy definitionBy definitionBy definition, velocity, velocity, velocity, velocity = distance/ time= distance/ time= distance/ time= distance/ time v = s/t..................v = s/t..................v = s/t..................v = s/t.................. (2)(2)(2)(2) In one coIn one coIn one coIn one complete vibration of the particle, distance travelled, s = mplete vibration of the particle, distance travelled, s = mplete vibration of the particle, distance travelled, s = mplete vibration of the particle, distance travelled, s = λ λ λ λ and time taken, t = Tand time taken, t = Tand time taken, t = Tand time taken, t = T From (2), v = From (2), v = From (2), v = From (2), v = λλλλ/T = /T = /T = /T = λ λ λ λ X1/T1/T1/T1/T Using (1),Using (1),Using (1),Using (1), we get we get we get we get

v = v = v = v = λ υλ υλ υλ υ .......... (.......... (.......... (.......... (3)3)3)3) Hence velocity of wave is the product of frequency and wavelength of Hence velocity of wave is the product of frequency and wavelength of Hence velocity of wave is the product of frequency and wavelength of Hence velocity of wave is the product of frequency and wavelength of the wave. This relation holds for transverse as well as longitudinal waves.the wave. This relation holds for transverse as well as longitudinal waves.the wave. This relation holds for transverse as well as longitudinal waves.the wave. This relation holds for transverse as well as longitudinal waves.

STANDING WAVES IN STRINGS AND

NORMAL MODES OF VIBRATION When a string under tension is set into vibrations, transverse harmonic When a string under tension is set into vibrations, transverse harmonic When a string under tension is set into vibrations, transverse harmonic When a string under tension is set into vibrations, transverse harmonic waves propagate along its length.waves propagate along its length.waves propagate along its length.waves propagate along its length. When the length of string is fixed,When the length of string is fixed,When the length of string is fixed,When the length of string is fixed, reflected waves will also exist.reflected waves will also exist.reflected waves will also exist.reflected waves will also exist. The incident and reflected waves will The incident and reflected waves will The incident and reflected waves will The incident and reflected waves will superimpose to produce transverse stationarsuperimpose to produce transverse stationarsuperimpose to produce transverse stationarsuperimpose to produce transverse stationary waves in the string.y waves in the string.y waves in the string.y waves in the string. The string will vibrate in such a way that the clamped points of the The string will vibrate in such a way that the clamped points of the The string will vibrate in such a way that the clamped points of the The string will vibrate in such a way that the clamped points of the string are nodes and the point of plucking is the antinode.string are nodes and the point of plucking is the antinode.string are nodes and the point of plucking is the antinode.string are nodes and the point of plucking is the antinode. Let a Let a Let a Let a harmonicharmonicharmonicharmonic wave be set up on a string of length L,wave be set up on a string of length L,wave be set up on a string of length L,wave be set up on a string of length L, fixed at the two fixed at the two fixed at the two fixed at the two ends x=0 and x=L.ends x=0 and x=L.ends x=0 and x=L.ends x=0 and x=L. TTTThis wave gets reflected from the two fixed ends of the his wave gets reflected from the two fixed ends of the his wave gets reflected from the two fixed ends of the his wave gets reflected from the two fixed ends of the string string string string continuouslycontinuouslycontinuouslycontinuously and as a result of superimposition of these waves,and as a result of superimposition of these waves,and as a result of superimposition of these waves,and as a result of superimposition of these waves, standing waves are formed on the string.standing waves are formed on the string.standing waves are formed on the string.standing waves are formed on the string. Let the wave pulse moving on the string from left to right be Let the wave pulse moving on the string from left to right be Let the wave pulse moving on the string from left to right be Let the wave pulse moving on the string from left to right be represented byrepresented byrepresented byrepresented by y1 = r sin y1 = r sin y1 = r sin y1 = r sin 2222π π π π (vt (vt (vt (vt ---- x)x)x)x) λλλλ WhereWhereWhereWhere the symbols have their usual meanings.the symbols have their usual meanings.the symbols have their usual meanings.the symbols have their usual meanings. Note that, here x is the Note that, here x is the Note that, here x is the Note that, here x is the distance from the origin in the direction of the wavedistance from the origin in the direction of the wavedistance from the origin in the direction of the wavedistance from the origin in the direction of the wave (from left to right).It (from left to right).It (from left to right).It (from left to right).It is often convenient to take the origin(x=0) at the interface (the site of is often convenient to take the origin(x=0) at the interface (the site of is often convenient to take the origin(x=0) at the interface (the site of is often convenient to take the origin(x=0) at the interface (the site of reflection),reflection),reflection),reflection), on the right fixed end of the string.on the right fixed end of the string.on the right fixed end of the string.on the right fixed end of the string. In that case,In that case,In that case,In that case, sign of x is sign of x is sign of x is sign of x is reversed because it is measured from the interface in a direction opposite to reversed because it is measured from the interface in a direction opposite to reversed because it is measured from the interface in a direction opposite to reversed because it is measured from the interface in a direction opposite to ththththe incident wave.e incident wave.e incident wave.e incident wave. The equation of incident wave may, therefore, be The equation of incident wave may, therefore, be The equation of incident wave may, therefore, be The equation of incident wave may, therefore, be written aswritten aswritten aswritten as y1 = r sin y1 = r sin y1 = r sin y1 = r sin 2222π π π π (vt + x).............(1)(vt + x).............(1)(vt + x).............(1)(vt + x).............(1) λλλλ As there is a phase change of As there is a phase change of As there is a phase change of As there is a phase change of ππππ radian on reflection at the fixed end of the radian on reflection at the fixed end of the radian on reflection at the fixed end of the radian on reflection at the fixed end of the

string,string,string,string, therefore,therefore,therefore,therefore, the reflected wave pulse travelling from right to left on the reflected wave pulse travelling from right to left on the reflected wave pulse travelling from right to left on the reflected wave pulse travelling from right to left on the string is represented bythe string is represented bythe string is represented bythe string is represented by y2 = r sin y2 = r sin y2 = r sin y2 = r sin [2[2[2[2π π π π ((((vt vt vt vt ---- x) + x) + x) + x) + ππππ ]]]] λλλλ = = = = - r sin r sin r sin r sin 2222π π π π (vt (vt (vt (vt ---- x)x)x)x)............ (............ (............ (............ (2)2)2)2) λλλλ According to superposition principle,According to superposition principle,According to superposition principle,According to superposition principle, the resultant displacement y at time t the resultant displacement y at time t the resultant displacement y at time t the resultant displacement y at time t and position x is given byand position x is given byand position x is given byand position x is given by y = y = y = y = y1 +y1 +y1 +y1 + y2y2y2y2 = r sin = r sin = r sin = r sin 2222π π π π (vt + x) (vt + x) (vt + x) (vt + x) - r sin r sin r sin r sin 2222π π π π (vt (vt (vt (vt ---- x)x)x)x) λ λ λ λ λλλλ = r [sin = r [sin = r [sin = r [sin 2222π π π π (vt + x) (vt + x) (vt + x) (vt + x) - sin sin sin sin 2222π π π π (vt (vt (vt (vt ---- x)].......(3)x)].......(3)x)].......(3)x)].......(3) λ λ λ λ λλλλ Using the relation,Using the relation,Using the relation,Using the relation, sin C sin C sin C sin C - sin D = 2 cos sin D = 2 cos sin D = 2 cos sin D = 2 cos C + D C + D C + D C + D sin sin sin sin C C C C ---- D D D D 2 2 2 2 2222 WeWeWeWe get,get,get,get,

y = 2 r cos y = 2 r cos y = 2 r cos y = 2 r cos 2 2 2 2 ππππ v t sin v t sin v t sin v t sin 2 2 2 2 ππππ x x x x λ λ λ λ λλλλ

……………………… (… (… (… (4) 4) 4) 4) AAAAs the arguments of trignometrical functions involved in (4) do not have s the arguments of trignometrical functions involved in (4) do not have s the arguments of trignometrical functions involved in (4) do not have s the arguments of trignometrical functions involved in (4) do not have the form (vt the form (vt the form (vt the form (vt ++++ x), therefore, it does not represent a moving harmonic wave.x), therefore, it does not represent a moving harmonic wave.x), therefore, it does not represent a moving harmonic wave.x), therefore, it does not represent a moving harmonic wave. Rather, it represents a new kind of waves called Rather, it represents a new kind of waves called Rather, it represents a new kind of waves called Rather, it represents a new kind of waves called standing or or or or stationary waves. At one end of the string, where x = 0At one end of the string, where x = 0At one end of the string, where x = 0At one end of the string, where x = 0 From (4),From (4),From (4),From (4), y = 2 r cos y = 2 r cos y = 2 r cos y = 2 r cos 2 2 2 2 ππππ vt sin vt sin vt sin vt sin 2 2 2 2 ππππ (0) = 0(0) = 0(0) = 0(0) = 0 λ λ λ λ λλλλ

At other end of the string, where x = LAt other end of the string, where x = LAt other end of the string, where x = LAt other end of the string, where x = L From (4),From (4),From (4),From (4), y = 2 r cos y = 2 r cos y = 2 r cos y = 2 r cos 2 2 2 2 ππππ vt sin vt sin vt sin vt sin 2 2 2 2 ππππ L ..........L ..........L ..........L .......... (5)(5)(5)(5) λ λ λ λ λλλλ As the other end of the string is fixed,As the other end of the string is fixed,As the other end of the string is fixed,As the other end of the string is fixed, ∴∴∴∴ y = 0, at this endy = 0, at this endy = 0, at this endy = 0, at this end For this, from (5),For this, from (5),For this, from (5),For this, from (5), sin sin sin sin 2 2 2 2 ππππ L = 0 = sin n L = 0 = sin n L = 0 = sin n L = 0 = sin n π,π,π,π, λλλλ where n = 1,2,3..........where n = 1,2,3..........where n = 1,2,3..........where n = 1,2,3.......... sin sin sin sin 2 2 2 2 ππππ L = n L = n L = n L = n ππππ λ λ λ λ

λ λ λ λ = = = = 2 L 2 L 2 L 2 L NNNN

.............(6) .............(6) .............(6) .............(6)

where n = 1where n = 1where n = 1where n = 1, 2, 3, 2, 3, 2, 3, 2, 3..... correspond to 1st..... correspond to 1st..... correspond to 1st..... correspond to 1st, 2nd,, 2nd,, 2nd,, 2nd, 3rd3rd3rd3rd..... normal modes of ..... normal modes of ..... normal modes of ..... normal modes of vibration of the string.vibration of the string.vibration of the string.vibration of the string. (i) First normal mode of vibration

Suppose Suppose Suppose Suppose λ1 λ1 λ1 λ1 is the wavelength of standing waves set up on the string is the wavelength of standing waves set up on the string is the wavelength of standing waves set up on the string is the wavelength of standing waves set up on the string corresponding to n = 1.corresponding to n = 1.corresponding to n = 1.corresponding to n = 1. From (6), From (6), From (6), From (6), λ1λ1λ1λ1 = = = = 2 L 2 L 2 L 2 L 1111 or L = or L = or L = or L = λ1λ1λ1λ1 2 2 2 2

The string vibrates as a whole in one segment, as shown in figure.The string vibrates as a whole in one segment, as shown in figure.The string vibrates as a whole in one segment, as shown in figure.The string vibrates as a whole in one segment, as shown in figure.

The frequency of vibration is given byThe frequency of vibration is given byThe frequency of vibration is given byThe frequency of vibration is given by υυυυ1 = 1 = 1 = 1 = v v v v = = = = vvvv ……….……….……….………. (a)(a)(a)(a) λλλλ1 1 1 1 2L2L2L2L As v =As v =As v =As v = √√√√T/mT/mT/mT/m wwwwhere There There There T is the tension in the string and m is the mass per unit length of is the tension in the string and m is the mass per unit length of is the tension in the string and m is the mass per unit length of is the tension in the string and m is the mass per unit length of the string.the string.the string.the string.

∴ υυυυ1 = 1 = 1 = 1 = 1 1 1 1 √√√√T T T T 2L m2L m2L m2L m

This normal mode of vibration is called This normal mode of vibration is called This normal mode of vibration is called This normal mode of vibration is called fufufufundamental ndamental ndamental ndamental mode. Themode. Themode. Themode. The frequency of vibration of string in this mode is minimum and is called frequency of vibration of string in this mode is minimum and is called frequency of vibration of string in this mode is minimum and is called frequency of vibration of string in this mode is minimum and is called fundamental frequency. The sound or note so produced is called The sound or note so produced is called The sound or note so produced is called The sound or note so produced is called fundamental note or first harmonic.

EXPERIMENT � OBJECTIVE-

To determine the frequency of AC mains by Melde’s experiment.To determine the frequency of AC mains by Melde’s experiment.To determine the frequency of AC mains by Melde’s experiment.To determine the frequency of AC mains by Melde’s experiment. � APPARATUS-

• Electrically maintained tuning forkElectrically maintained tuning forkElectrically maintained tuning forkElectrically maintained tuning fork • A stand with clamp and pulleyA stand with clamp and pulleyA stand with clamp and pulleyA stand with clamp and pulley • A light weight panA light weight panA light weight panA light weight pan • A weight boxA weight boxA weight boxA weight box • BalanceBalanceBalanceBalance • A battery with eliminator and connecting wiresA battery with eliminator and connecting wiresA battery with eliminator and connecting wiresA battery with eliminator and connecting wires

� THEORY- A string can be set into vibrations by means of an electrically A string can be set into vibrations by means of an electrically A string can be set into vibrations by means of an electrically A string can be set into vibrations by means of an electrically maintained tuning fork,maintained tuning fork,maintained tuning fork,maintained tuning fork, thereby producing stationary wavesthereby producing stationary wavesthereby producing stationary wavesthereby producing stationary waves due to due to due to due to reflection of waves at the pulley. The end of the pulley where it touches reflection of waves at the pulley. The end of the pulley where it touches reflection of waves at the pulley. The end of the pulley where it touches reflection of waves at the pulley. The end of the pulley where it touches the pulley and the position where it is fixed to thethe pulley and the position where it is fixed to thethe pulley and the position where it is fixed to thethe pulley and the position where it is fixed to the prong of tuning fork.prong of tuning fork.prong of tuning fork.prong of tuning fork.

(i)For the transverse arrangement,(i)For the transverse arrangement,(i)For the transverse arrangement,(i)For the transverse arrangement, the frequency is given bythe frequency is given bythe frequency is given bythe frequency is given by n = n = n = n = 1111 √√√√T T T T 2L 2L 2L 2L mmmm where ‘L’ is the length of thread in fundamental modes of vibrationswhere ‘L’ is the length of thread in fundamental modes of vibrationswhere ‘L’ is the length of thread in fundamental modes of vibrationswhere ‘L’ is the length of thread in fundamental modes of vibrations, , , , ‘‘‘‘ T ’T ’T ’T ’ is the tension applied to the thread and ‘m’ is the mass per unit is the tension applied to the thread and ‘m’ is the mass per unit is the tension applied to the thread and ‘m’ is the mass per unit is the tension applied to the thread and ‘m’ is the mass per unit length of thread.length of thread.length of thread.length of thread. If ‘p’ loops are formed in the length ‘L’ of the threadIf ‘p’ loops are formed in the length ‘L’ of the threadIf ‘p’ loops are formed in the length ‘L’ of the threadIf ‘p’ loops are formed in the length ‘L’ of the thread,,,, thenthenthenthen

n = n = n = n = pppp √√√√T T T T 2L m2L m2L m2L m

(ii)For the longitudinal arrangement,(ii)For the longitudinal arrangement,(ii)For the longitudinal arrangement,(ii)For the longitudinal arrangement, when ‘p’ loops are formed,when ‘p’ loops are formed,when ‘p’ loops are formed,when ‘p’ loops are formed, the the the the frequency is given byfrequency is given byfrequency is given byfrequency is given by

n = n = n = n = pppp √√√√T T T T L mL mL mL m

� PROCEDURE-

• Find the weight of pan P and arrange the apparatus as shown in Find the weight of pan P and arrange the apparatus as shown in Find the weight of pan P and arrange the apparatus as shown in Find the weight of pan P and arrange the apparatus as shown in figure.figure.figure.figure.

• Place a load of 4 Place a load of 4 Place a load of 4 Place a load of 4 TTTTo 5 gm in the pan attached to the o 5 gm in the pan attached to the o 5 gm in the pan attached to the o 5 gm in the pan attached to the end ofend ofend ofend of the string the string the string the string passing over the passing over the passing over the passing over the pulleypulleypulleypulley.... Excite the tuning fork by switching on the Excite the tuning fork by switching on the Excite the tuning fork by switching on the Excite the tuning fork by switching on the power supply.power supply.power supply.power supply.

• Adjust the position of the pulley so that the string is set into resonant Adjust the position of the pulley so that the string is set into resonant Adjust the position of the pulley so that the string is set into resonant Adjust the position of the pulley so that the string is set into resonant vibrations and well defined loops are vibrations and well defined loops are vibrations and well defined loops are vibrations and well defined loops are obtained. Ifobtained. Ifobtained. Ifobtained. If necessary, adjustnecessary, adjustnecessary, adjustnecessary, adjust the tensions by adding weights in the pan slowly and the tensions by adding weights in the pan slowly and the tensions by adding weights in the pan slowly and the tensions by adding weights in the pan slowly and gradually. Forgradually. Forgradually. Forgradually. For finer finer finer finer adjustment, addadjustment, addadjustment, addadjustment, add milligram weight so that nodes are reduced to milligram weight so that nodes are reduced to milligram weight so that nodes are reduced to milligram weight so that nodes are reduced to points.points.points.points.

• Measure the length of say 4 loops formed in the middle part of the Measure the length of say 4 loops formed in the middle part of the Measure the length of say 4 loops formed in the middle part of the Measure the length of say 4 loops formed in the middle part of the string. Ifstring. Ifstring. Ifstring. If ‘L’ is the distance in which 4 loops are ‘L’ is the distance in which 4 loops are ‘L’ is the distance in which 4 loops are ‘L’ is the distance in which 4 loops are formed, thenformed, thenformed, thenformed, then distance between two consecutive nodes is L/4.distance between two consecutive nodes is L/4.distance between two consecutive nodes is L/4.distance between two consecutive nodes is L/4.

• Note down the weight placed in the pan and calculate the tension T.Note down the weight placed in the pan and calculate the tension T.Note down the weight placed in the pan and calculate the tension T.Note down the weight placed in the pan and calculate the tension T.

Tension, TTension, TTension, TTension, T= (= (= (= (wt. in the pan + wt. of panwt. in the pan + wt. of panwt. in the pan + wt. of panwt. in the pan + wt. of pan)))) gggg • Repeat the experiment Repeat the experiment Repeat the experiment Repeat the experiment twinetwinetwinetwine by changing the weight in the pan in by changing the weight in the pan in by changing the weight in the pan in by changing the weight in the pan in

steps of one gram ansteps of one gram ansteps of one gram ansteps of one gram and altering the position of the pulley each time to d altering the position of the pulley each time to d altering the position of the pulley each time to d altering the position of the pulley each time to get well defined loops.get well defined loops.get well defined loops.get well defined loops.

• Measure one Measure one Measure one Measure one metermetermetermeter length of the thread and find its mass to find the length of the thread and find its mass to find the length of the thread and find its mass to find the length of the thread and find its mass to find the value of m,value of m,value of m,value of m, the mass produced per unit length.the mass produced per unit length.the mass produced per unit length.the mass produced per unit length.

� OBSERVATIONS AND

CALCULATIONS- For longitudinal arrangementFor longitudinal arrangementFor longitudinal arrangementFor longitudinal arrangement WeightWeightWeightWeight No. of No. of No. of No. of

loopsloopsloopsloops Length of Length of Length of Length of

threadthreadthreadthread Length of Length of Length of Length of each loopeach loopeach loopeach loop

TensionTensionTensionTension nnnn

20202020 4444 152152152152 38383838 36363636 45.545.545.545.5

30303030 4444 143143143143 35.7535.7535.7535.75 46464646 54545454

40404040 3333 130130130130 43.343.343.343.3 56565656 49.349.349.349.3

Mean frequency=49.6 vib/secMean frequency=49.6 vib/secMean frequency=49.6 vib/secMean frequency=49.6 vib/sec For transverse arrangementFor transverse arrangementFor transverse arrangementFor transverse arrangement

WeightWeightWeightWeight No. of No. of No. of No. of loopsloopsloopsloops

Length of Length of Length of Length of threadthreadthreadthread

Length of Length of Length of Length of each loopeach loopeach loopeach loop

TensionTensionTensionTension nnnn

40404040 7777 157157157157 21.521.521.521.5 56565656 49.749.749.749.7 50505050 6666 145145145145 24.124.124.124.1 66666666 48.148.148.148.1 60606060 5555 137137137137 27.427.427.427.4 76767676 45.445.445.445.4 Mean frequency=47.7 vib/secMean frequency=47.7 vib/secMean frequency=47.7 vib/secMean frequency=47.7 vib/sec Mass of the pan, W=……… kgMass of the pan, W=……… kgMass of the pan, W=……… kgMass of the pan, W=……… kg

Mass per Mass per Mass per Mass per metermetermetermeter of thread,of thread,of thread,of thread, m=……… kgm=……… kgm=……… kgm=……… kg For transverse arrangement,For transverse arrangement,For transverse arrangement,For transverse arrangement,

n = n = n = n = 1111 √√√√T T T T 2L m2L m2L m2L m For longitudinal arrangement,For longitudinal arrangement,For longitudinal arrangement,For longitudinal arrangement,

n = n = n = n = 1111 √√√√T T T T L mL mL mL m Mean frequency, n=………… vib/sec.Mean frequency, n=………… vib/sec.Mean frequency, n=………… vib/sec.Mean frequency, n=………… vib/sec. � PRECAUTIONS-

• The thread should be uniform and inextensible.The thread should be uniform and inextensible.The thread should be uniform and inextensible.The thread should be uniform and inextensible. • Well defined loops should be obtained by adjusting the tension with Well defined loops should be obtained by adjusting the tension with Well defined loops should be obtained by adjusting the tension with Well defined loops should be obtained by adjusting the tension with

milligrammilligrammilligrammilligram weights.weights.weights.weights. • Frictions in the pulley should be least possible.Frictions in the pulley should be least possible.Frictions in the pulley should be least possible.Frictions in the pulley should be least possible.