p ph h y y s s i i c c s s p pr r o o j j e e c ct t r re e p po o r r t t
m me e l l d d e e ’ ’ s s e e x xp p e e r r i i m m e e n n t t
Submitted to: Mr. rohit verma
Submitted by: group 27,b3
TYPES OF WAVE MOTION
The mechanical waves are of two types. • Transverse wave motion • Longitudinal wave motion Transverse wave motion-
A transverse wave motion is that wave motion, in which which individual individual particles of the medium execute simple harmonic motion about their mean position in a direction perpendicular to the direction of propagation of wave motion.
For example example- - - (i) (i) Movement Movement of string of a sitar or violin (ii) (ii ) Movement Movement of membrane of a tabla (iii) (iii ) Movement Movement of a kink on a rope
Waves set up on the surface of water are a combination of transverse waves and longitudinal waves. Light waves and all other electro electro- -magnetic -magnetic waves are also transverse waves. A A transverse wave travels through a medium in the form of crests and troughs. A crest is a portion of the medium which is raised temporarily above the normal position of rest of the particles of the medium, when when aa transverse transverse wave passes through it. The The centre of crest is the position of maximum cc entre entre displacement in the positive direction. A trough is a portion of the medium which is depressed temporarily below the normal position of rest of the particles of the medium, when a transverse wave passes through it. The centre of trough is the position of maximum displacement in the negative dir direction. ection. The distance between two consecutive crests or two consecutive troughs is called wavelength of the wave. It is represented by λ λ Thus AC = BD = λ λ For the propagation of mechanical waves, the material medium must possess the following characteristics: (i) Elasticity , so that particles can return to their mean position, after after having been disturbed. (ii) Inertia , so that particles can store energy and overshoot their mean position.
SOME TERMS CONNECTED WITH WAVE MOTION Wavelength- Wavelength of a wave is the length of one wave. It
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is equal to the distance travelled by the wave during the time; any one the time; particle of the medium completes one vibration about its mean position. We may also define wavelength as the distance between any two nearest particles of the medium, vibrating in the same phase. phase. As stated already transverse wave motion, λ = distance between between centers centers of two consecutive crests or distance between between centers centers of two consecutive troughs. Also, wavelength can be taken as the distance in which one crest and one trough are contained. Similarly, in a longitudinal wave motion, λ = distance between the the centers centers of two consecutive compressions or distance between two consecutive rarefactions.
Also, wavelength can be taken as the distance in which one compression and one rarefaction are contained.
Frequency-Frequency of vibration of a particle is defined as the
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number of vibrations completed by particle in one second. As As one vibration is equivalent to one wavelength, therefore, we may define frequency of a wave as the number of complete wavelengths transversed υ. transverse d by the wave in one second. It is represented by υ •
Time period-Time period of vibration of a particle is defined as
the time taken by the particle to complete one vibration about its mean position. As As one vibration is equivalent to one wavelength, therefore, time period of a wave is equal to time taken by the wave to travel a distance equal to one wavelength. It is represented by by T T T.
RELATION BETWEEN υ AND T
By definition, Time for completing v vibrations = 1 sec Time for completing 1 vibration = 1/ υ υ sec i.e. T = 1/ υ υ or υ υ = 1/T or υ υT =1
…………. (1)
RELATION BETWEEN VELOCITY, FREQUENCY AND WAVELENGTH OF A WAVE
Suppose υ υ = frequency of a wave T = time period of the wave λ = wavelength of the wave v = velocity of the wave. By definition definition,, velocity velocity = distance/ time v = s/t.................. (2) In one complete co mplete vibration of the particle, distance travelled, s = λ and time taken, t = T From (2), v = λ 1/T λ /T = λ X 1/T Using (1), we get v = λ υ
.......... ( (3) 3)
Hence velocity of wave is the product of frequency and wavelength of 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 waves propagate along its length. When the length of string is fixed, reflected waves will also exist. The incident and reflected waves will superimpose to produce transverse stationary stationar y waves in the string. 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. Let aa harmonic fixed at the two harmonic wave be set up on a string of length L, fixed ends x=0 and x=L. T This This wave gets reflected from the two fixed ends of the string string continuously continuously and as a result of superimposition of these waves, standing waves are formed on the string. Let the wave pulse moving on the string from left to right be represented by y1 = r sin sin 2 2π π (vt (vt -- x) λ λ
Where Where the symbols have their usual meanings. Note that, here x is the distance from the origin in the direction of the wave wave (from left to right).It is often convenient to take the origin(x=0) at the interface (the site of reflection), on the right fixed end of the string. In that case, sign of x is reversed because it is measured from the interface in a direction opposite to the th e incident wave. The equation of incident wave may, therefore, be written as y1 = r sin sin 2 2π π (vt + x).............(1) λ λ
As there is a phase change of π π radian on reflection at the fixed end of the
string, therefore, the reflected wave pulse travelling from right to left on the string is represented by π (vt y2 = r sin π ] sin [2 π (( vt vt - - x) + π ] =
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λ λ r sin sin 2 x)............ ............ ( (2) 2) 2π π (vt (vt -- x) λ λ
According to superposition principle, the resultant displacement y at time t and position x is given by y == y1 y1 + + y2 y2 = r sin sin 2 sin 2 2π π (vt + x) r sin 2π π (vt (vt -- x) -
λ λ = r [sin [sin 2 2π π (vt + x) λ λ
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λ λ sin sin 2 2π π (vt (vt -- x)].......(3) λ λ
Using the relation, sin C sin D = 2 cos D sin cos C + D sin CC - - D 22 2 2 -
We get, We get, y = 2 r cos sin 2 π π v t sin π x cos 2 π λ λ
λ λ
……… 4) ……… ( (4) As A A As the arguments of trignometrical functions involved in (4) do not have the form (vt + x), therefore, it does not represent a moving harmonic wave. Rather, it represents a new kind of waves called standing or stationary waves.
At one end of the string, where x = 0 From (4), y = 2 r cos sin 2 π π vt sin π (0) = 0 cos 2 π λ λ
λ λ
At other end of the string, where x = L From (4), y = 2 r cos sin 2 π π vt sin π L .......... (5) cos 2 π λ λ
λ λ
As the other end of the string is fixed, ∴ yy = 0, at this end end For this, from (5), sin π L = 0 = sin n π, sin 2 π λ λ
where n = 1,2,3.......... sin π L = n π π sin 2 π λ λ = = 2 L
N N
.............(6)
where n = 1, ..... correspond to 1st, 3rd ..... normal modes of 1, 2, 33..... 1st , 2nd, 3rd..... 3rd vibration of the string. (i) First normal mode of vibration
Suppose λ1 is the wavelength of standing waves set up on the string corresponding to n = 1. From (6), λ1 == 2 L 1 or L = λ1 2
The string vibrates as a whole in one segment, as shown in figure.
The frequency of vibration is given by υ υ1 = = vv == vv ………. (a) λ1 λ 1 2L λ As v == √T/m where w here w TT is the tension in the string and m is the mass per unit length of the string. ∴
υ T υ1 = = 1 √ T
2L m
This normal mode of vibration is called fu called fundamental fu ndamental ndamental mode. The frequency of vibration of string in this mode is minimum and is called fundamental frequency. 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. APPARATUS-
Electrically maintained tuning fork • A stand with clamp and pulley • A light weight pan • A weight box • Balance • A battery with eliminator and connecting wires •
THEORY-
A string can be set into vibrations by means of an electrically maintained tuning fork, thereby producing stationary waves waves due to reflection of waves at the pulley. The end of the pulley where it touches the pulley and the position where it is fixed to the prong of tuning fork. the prong (i)For the transverse arrangement, the frequency is given by n == 1 √ T T 2L 2L m m where ‘L’ is the length of thread in fundamental modes of vibrations,, vibrations , ‘‘ T ’’ is the tension applied to the thread and ‘m’ is the mass per unit length of thread. If ‘p’ loops are formed in the length ‘L’ of the thread thread,, then n == pp T p √ T 2L m (ii)For the longitudinal arrangement, when ‘p’ loops are formed, the frequency is given by
n == pp T p √ T L m
PROCEDURE-
Find the weight of pan P and arrange the apparatus as shown in figure. • Place a load of 4 4 T To of the string To 5 gm in the pan attached to the the end of passing over the pulley . Excite the tuning fork by switching on the the pulley. pulley power supply. • Adjust the position of the pulley so that the string is set into resonant vibrations and well defined loops are If necessary, adjust are obtained. If the tensions by adding weights in the pan slowly and and gradually. gradually. For finer add milligram weight so that nodes are reduced to finer adjustment, add points. • Measure the length of say 4 loops formed in the middle part of the string. If are formed, formed, then If ‘L’ is the distance in which 4 loops are then distance between two consecutive nodes is L/4. • Note down the weight placed in the pan and calculate the tension T. •
Tension, T= pan) )) g T = (wt. ( wt. in the pan + wt. of pan g g g twine by changing the weight in the pan in • Repeat the experiment experiment twine steps of one gram and an d altering the position of the pulley each time to get well defined loops. one meter meter length of the thread and find its mass to find the • Measure one value of m, the mass produced per unit length. OBSERVATIONS AND CALCULATIONS-
For longitudinal arrangement Weight Weight
No. of of Length of loops thread
20 44 152 152 20 30 44 143 30 40 3 130 130 40 Mean frequency=49.6 vib/sec
Length of Tension Tension each loop 38 38 35.75 35.75 43.3
36 36 46 46 56 56
n n 45.5 54 49.3
For transverse arrangement Weight Weight
No. of of Length of loops thread 40 77 157 40 157 50 66 145 50 145 60 55 137 60 137 Mean frequency=47.7 vib/sec
Mass of the pan, W=……… kg
Length of Tension Tension n n each loop 21.5 21.5 56 56 49.7 24.1 66 66 48.1 27.4 27.4 76 76 45.4
Mass per per meter meter of thread, m=……… kg For transverse arrangement, n == 1 √ T T 2L m For longitudinal arrangement, n == 1 √T L m Mean frequency, n=………… vib/sec. PRECAUTIONS• •
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The thread should be uniform and inextensible. Well defined loops should be obtained by adjusting the tension with milligram milligram weights. Frictions in the pulley should be least possible.
