Lab Mannual Melde's Experiment

physics project report

melde’s e xp e ri m e n 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 motionA transverse wave motion is that wave motion, in which 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 exampleexample(i) Movement of string of a sitar or violin (ii) (ii) Movement of membrane of a tabla (iii) (iii) 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 electroelectro-magnetic waves are also transverse waves. 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 a transverse wave passes through it. The centre of crest is the position of maximum 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 direction. direction. The distance between two consecutive crests or two consecutive troughs is called wavelength of the wave. It is represented by λ Thus AC = BD = 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 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

is equal to the distance travelled by the wave during the time; any one 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. As stated already transverse wave motion, λ = distance between centers of two consecutive crests or distance between 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 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

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

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

…………. (1)

RELATION BETWEEN VELOCITY, FREQUENCY AND WAVELENGTH OF A WAVE Suppose 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 = distance/ time v = s/t.................. (2) In one complete complete vibration of the particle, distance travelled, s = λ and time taken, t = T 1/T From (2), v = λ/T = λ X1/T Using (1), we get v=λυ

.......... (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 stationary 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 a harmonic wave be set up on a string of length L, fixed at the two ends x=0 and x=L. This wave gets reflected from the two fixed ends of the string 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 2π (vt - x)

λ

Where the symbols have their usual meanings. Note that, here x is the distance from the origin in the direction of the 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 the incident wave. The equation of incident wave may, therefore, be written as y1 = r sin 2π (vt + x).............(1)

λ As there is a phase change of 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 y2 = r sin [2π (vt - x) + π ]

λ = - r sin 2π (vt - x)............ x)............ (2) (2) λ

According to superposition principle, the resultant displacement y at time t and position x is given by y = y1 + y2 = r sin 2π (vt + x) - r sin 2π (vt - x)

λ

λ = r [sin 2π (vt + x) - sin 2π (vt - x)].......(3) λ λ

Using the relation, sin C - sin D = 2 cos C + D sin C - D 2 2 We get,

y = 2 r cos 2 π v t sin 2 π x

λ

λ

……… ……… (4) (4) 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 2 π vt sin 2 π (0) = 0

λ

λ

At other end of the string, where x = L From (4), y = 2 r cos 2 π vt sin 2 π L .......... (5)

λ

λ

As the other end of the string is fixed, y = 0, at this end ∴ For this, from (5), sin 2 π L = 0 = sin n π,

λ where n = 1,2,3.......... sin 2 π L = n π

λ λ=2L N

.............(6) where n = 1, 1, 2, 3..... 3..... correspond to 1st, 1st, 2nd, 3rd..... 3rd..... normal modes of 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 υ1 = v = v ………. (a) λ1 2L As v = √T/m where T is the tension in the string and m is the mass per unit length of the string. ∴

υ1 = 1 √T 2L

m

This normal mode of vibration is called fundamental fundamental 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 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. (i)For the transverse arrangement, the frequency is given by n = 1 √T 2L 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 = p √T 2L m (ii)For the longitudinal arrangement, when ‘p’ loops are formed, the frequency is given by

n = p √T L m


PROCEDURE-

• Find the weight of pan P and arrange the apparatus as shown in figure. • Place a load of 4 To 5 gm in the pan attached to the end of the string passing over the pulley. pulley. Excite the tuning fork by switching on the power supply. • Adjust the position of the pulley so that the string is set into resonant vibrations and well defined loops are obtained. If necessary, adjust the tensions by adding weights in the pan slowly and gradually. For finer adjustment, add milligram weight so that nodes are reduced to points. • Measure the length of say 4 loops formed in the middle part of the string. If ‘L’ is the distance in which 4 loops are formed, then distance between two consecutive nodes is L/4. • Note down the weight placed in the pan and calculate the tension T.

Tension, T= T= (wt. (wt. in the pan + wt. of pan) pan) g • Repeat the experiment twine by changing the weight in the pan in steps of one gram and and altering the position of the pulley each time to get well defined loops. • Measure one meter length of the thread and find its mass to find the value of m, the mass produced per unit length.




OBSERVATIONS AND CALCULATIONS-

For longitudinal arrangement Weight

No. of loops

Length of thread

Length of each loop

Tension

n

20 30

4 4

152 143

38 35.75

36 46

45.5 54

40

3

130

43.3

56

49.3

Length of each loop 21.5 24.1 27.4

Tension

n

56 66 76

49.7 48.1 45.4

Mean frequency=49.6 vib/sec For transverse arrangement Weight

No. of Length of loops thread 40 7 157 50 6 145 60 5 137 Mean frequency=47.7 vib/sec

Mass of the pan, W=……… kg

Mass per meter of thread, m=……… kg For transverse arrangement, n = 1 √T 2L m For longitudinal arrangement, n = 1 √T L m Mean frequency, n=………… vib/sec.


PRECAUTIONS-

• The thread should be uniform and inextensible. • Well defined loops should be obtained by adjusting the tension with milligram weights. • Frictions in the pulley should be least possible.