EXPERIMENT #25

YOUNG'S MODULUSEXEERIMENTAT FINDING FROM THE SPEED OF SOUND

. , i : " . . ; . . . ^ . . , : i ) . : : . , :

. '

Theory, DefinitionsTheoretical background concerning elasticity of solids is described in

Experiment #7.lf we can determine the speed of sound c in a thin homogenous rod with known

density p, then according to Eq. (7.6) we can calculate Young's modulus as

E=pc2

Tfie speed of sound in a particular rod can be easilyKundfs tube (see Fig. 25-1).

L.gngitudinal standing wave motion of sound in Kundt's tube is visible by meansof light cork particles uniformly placed in the glass tube. When the rod is rubbed, itsvibrations are transmitted into the glass tube and the light cork particles movetogether with the air inside the tube --*" ,rn onr"*" tne Inape of *"u" motion inthe tube - both the nodes and crests (antinodes).

The frequency of vibrations in the rod f has to be equal to the frequency of thevibrations of sound through the air f in the tube, that means

f = f (25.21

Fig. 25-{. Kundt's tube

Standing waves in air and in the tube can be described by the known formulas

(25.1)

determined by means of

I

Hzi-r)$

(25.3)

f are related(quantities c, .'1, f are related to the measured rod, and quantities C , A',to the corresponding quantities in the tube)

Let us substitute this into formula (25.2), then with respect to relation (25.1) weobtain

X=(z;-r)I

125.41

What should be substituted for the wavelength ,l of sound propagation in the rod?The wavelength depends on the way the rod is being fixed. lf it is fixed at one

point in the center of the rod, then i is two times the rod length / .'

(25.5a)

Of course, the rod can be fixed in two points, each at a distance of Ta I from theends of the rod, then

I =4. (26.5b)

(25.6)

where the speed of sound in the air c'is dependent on the air temperature accordingto the relation

c' = 1344.3 + 0.62 (fl -20)l ffi.s-1 , (25.71

where f is the air temperature in degrees of CelsiusUsing the method of successive measurements of the distance yl of two

srccessi-ve nodes (antinodes) we get the numerical value of 1,,'12 (being the slope ofa straight line dependence) in the form

Both relations (25.5a), (25.5b) follow from the different wari'bform of the- qoundwave in the rod, thai means irom different positions of its nodes.

' - ...i j. i...

-'t '.."-'ih*

; fi; case described by equation (25.5a), toimul" (25:4i:can b-e'iSwriite'n.in the next form

E = e[j'[*)]'

,_ A' -

2

i=o (r,#), ,

(25.8)

Note i: . '

Let us assume the case where the rod is fixed in its center. Then, the harmonicwave with different frequencies can arise in the rod - that with the lower frequency iscalled fundamental harmonic wave, all other are called higher harmonic waves'

Of course, the length / of the rod is related to the individual wavelength ;liaccording to the relation

A' ,2g-= -> l / ,2 n (n+1) f ; ' '

t = (2r-rlf ,

where the order number of the higher wavelength is an integer number

k = 1 ,2 ,3 , . . .

A similar condition describes the relation between the length of glass tube /' andall possible harmonic wave into this tube (see also Fig. 25-1)

l '= (2k-1)&2

Objectives of the Measurement ', ' l1. Find the wavelength of the longitudinal standing wave in the air column in

glass tube.2. Determine Young's modulus,E and its uncertainty.3. Try to explain formula (25.9) for the error of Young's modulus calculation.4. Compare your result with the accepted value of E for a given rnaterial of a

(made either of steel or cooper).

the

rod

Procedure of the Measurement1. Measure all quantities in Eq. (25.6). Determine the air temperature.2. Repeat measurements of wavelength of sound in air (distance between

successive nodes) using method of successive measurements. Use a similarway of measurement processing shown in the next table

2y, = Z(y, - bi )2 =

Estimate the error of I and calculate the error of l'.Calculate the speed of sound in air from measured temperature of surroundingair.Calculate the true error of your measurement {as an absolute value of thedifference of the measured value and accepted value of Young's modulus), andcornpare it with the caicuiateci error. Then, you can decide, it you have taken aiiuncertainties into account, or not. $ummarise it into your conclusion.

?

4.

5 .

1 2 3 4 5 6 7 I I 10

Yilcrnl

b ilcml

Yi-b ilcml

v)

NoteYou can measure Young's modulus

data are valid for both materialsdensity of brassdensity of steelYoung's modulus of brass ' -

Young's modulus of steel

; : , . I . . : j t '

of either brass or steel rod. The following

,obrass =

Psteel =p=f ;=

(8 410 * 50) kg.m-u(7 Bso;t,.s0) kg.m-32.1x1011 Pa1.0 x 1011 Pa

Accuracy of the MeasurementFinalLxpression for calculating of Young's modulus (26.5) gives the expression

for its error in the next form

K(E) = Er,{F} = E[r,(p)+2r,(c')+2r,(/)+2r,( ' l ' ) ] (25'9]- , i i

where it is'possible to estimate the error of the speerl of sound in air as

x(c] = 0"62 {r(f)} ,

and k(,1) is given bY

i l k(A') '= ' ' , , [

# ' "(26.101

Glossary

speed rof' sound ProPagationKundt's tubelongitudinal standing wavenodescrests (antinodes)location of nodes

rychlost Siieni.zvukuKundtova trubicepod6ln6 stojatS vlnauzlykmitnypoloha uzlft

Student's Notes

; ' 1

Z a . i . . :

, l

t

96