8. STATIONARY WAVES. Formula :

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1 8. SAIONARY WAVES. A sooeter wire of egth 0.5 is stretched by a weight of 5 kg. he fudaeta frequecy of ibratio is 00 Hz. Deterie the iear desity of ateria of wire. 0.5 Mg N 00 Hz? Forua : Soutio : Squarig both sides, we get (00) kg/ 9 9 ( 00) kg/. A sooeter wire 00 c og produces a resoace with a tuig fork. Whe its egth is decreased by 0 c, 8 beats per secod are heard. Fid the frequecy of tuig fork. 00 c c N ~ 8s? Forua : Soutio : Sice, ad > the > Hz 7 Hz 3. wo wires of the sae ateria ad haig the sae radius hae their fudaeta frequecies i the ratio :, ad tesio i the ratio : 8 copare ratio of their egths. 8 Statioary Waes

2 .. 0 MAHESH UORIAS SCIENCE Forua : Soutio :? For secod case,... (i)... (ii) [ is costat because both wires are ade of sae ateria] Diide (i) by (ii) :. A wire, og ad weighig g, wi be i resoace with a frequecy of 300 Hz. Fid tesio o stretchig the wire. M g 0 3 kg 300 Hz? Forua : Soutio : M Now, kg/ (600) N 5. A stretched sooeter wire is i uiso with a tuig fork, whe the egth is icreased by %, the uber of beats heard per secod is 6. fid the frequecy of the fork..0.0, N 6s? Statioary Waes

3 MAHESH UORIAS SCIENCE.. Forua : Soutio : Sice > the > 6 Hz 6 For two wires of sae ateria But, 6 ad.0 ( 6) (.0 ) ( 6) (.0) Hz 6. he speed of a traserse wae aog a uifor eta wire, whe it is uder a tesio of 000 g wt. is 68 /s. If the desity of eta is 7900 kg/ 3. Fid the area of cross sectio of the wire. 000 g wt kg wt. 9.8 N 9.8 N 68 /s ρ 7900 kg/ 3 A? Forua : Soutio : Sice ass of the wire, M Vρ Aρ Aso, M Aρ Now, A A Aρ Aρ ρ ( ) Aρ A A traserse wae is produced o a streched strig 0.7 og ad fixed at its eds. Fid speed of traserse wae, whe it ibrates, eittig the secod oertoe of frequecy 300 Hz Hz? Forua : Soutio : I d oertoe, 3 oops are fored 3 3 Statioary Waes

4 .. MAHESH UORIAS SCIENCE Now, 8. A uifor wire uder tesio, is fixed at its eds. If the ratio of tesio i the wire to the square of its egth is 360 dye/c ad fudaeta frequecy of ibratio of wire is 300 Hz. Fid its iear desity. 360 dye/c 300 Hz? Forua : Soutio : /s.. ( 300) g/c 0 kg/ A wire is i uiso with a fork of frequecy 50 Hz, whe streched by a weight hagig erticay. O iersig the weight i water, the wire produces te beats per secod with the sae fork. Cacuate desity of ateria of weight, Whe wire is stretched by a weight hagig erticay, 50 Hz, Frequecy of wire whe the weight is iersed i water producig 0 beats per secod Hz Forua : ρ w g/cc ρ? Soutio : ρ ρ ρ ρ 50 0 ρ ρ 5 ρ ρ Squarig both sides, 65 ρ 576 ρ 65 (ρ ) 576 ρ 65 ρ 576 ρ 65 9 ρ 65 ρ 65 9 ρ.76 g/c 3 Statioary Waes

5 MAHESH UORIAS SCIENCE wo sipe haroic progressie waes are represeted by x y si π 00t 60 c ad y si π 00t + x 60 c. he waes cobie to fro a statioary wae. Fid : i) apitude at atiode ii) distace betwee adjacet ode ad atiode iii) oop egth i) wae eocity y si π x 00t c y si π x 00t + c i) R? ii)? iii)? i)? Forua : y R si πt where, R A cos π x Soutio : i) Resutatat equatio of wae is gie by y y + y x si π 00t + si π x 00t + y cos π x si π (00 t)...(i) Coparig aboe equatio with, y R si (πt) We get, cos πx R But, R π cos πr 60 c Apitude at atiode is axiu aue of R. πx i.e, R is axiu whe cos R c ii) 60 c 60 5 c Distace betwee successie ode ad atiode 5 c iii) egth of oop 30 c i) wae eocity 60 Fro equatio (i), we get, 00 Hz 60 c c/s 60 /s y si π (00 t)cos π x 60 Statioary Waes

6 .. MAHESH UORIAS SCIENCE. he equatio of a stadig wae is gie by y 0.0 cos (πx) si (00 π t). Fid the apitude of either wae iterferig, waeegth, tie period, frequecy ad wae eocity of iterferig waes. y 0.0 cos (πx) si (00 πt) A????? Forua : y R si πt where, R A cos π x Cacuatio : y 0.0 cos (πx) si (00 π t) y 0.0 cos π x si[π (50)t] Coparig the gie equatio with, y R si πt where, R A cos π x We get, A apitudeof iterferig waes 0.0 Waeegth of iterferig waes Frequecy of iterferig waes 50 Hz ie period iterferig waes Veocity of iterferig waes /s. I Mede s experiet, fid weight added i the pa whe uber of oops o the strig chages fro to. If iitia tesio o the strig is 960 dye ad ass of the pa i oe gra. p Forua : Soutio : p M 0 g (M 0 + M )g 960 dye (M 0 + M )g M? p p p p p p ( M 0 + M ) g ( ) ( ) ( ) 6 M + M g But (M 0 + M )g (M 0 + M )g 960 ( + M ) (M + ) M + 8 M 8 M 7 g wt Statioary Waes

7 MAHESH UORIAS SCIENCE I Mede s experiet, fork was arraged i parae positio ad 6 oops were fored aog a egth of 7. whe stretched by a weight of 0 g. If ass of the strig is. 0 g, fid the freqeucy of tuig fork. p 6 7. M 0 g kg M. 0 g. 0 5 kg N? Forua : Soutio : P P For parae positio, frequecy of fork is gie by N N N Now,.P P M kg/ Substitutig the aues i (i), we get, N N (i) N N Hz. Fid the frequecy of fifth oertoe of a air cou ibratig i a pipe cosed at oe ed, egth of pipe is.0 c ad speed of soud i air at roo teperature is /s. [Ier diaeter of pipe is 3.5 c].0c /s d 3.5 c Frequecy of fifth oertoe 5? Forua : Fudaeta frequecy i air cou cosed at oe ed is gie by V Soutio : d Now, V ( 0.35) Hz P th oertoe is gie by p (p + ) Frequecey of fifth oertoe is, 5 ( 5 + ) Hz Hz Statioary Waes 70.

8 .. 6 MAHESH UORIAS SCIENCE 5. wo orga pipes, ope at both eds, are souded together ad 5 beats are heard per secod. he egth of shorter pipe is 0.5. fid the egth of the otherpipe. ( eocity of soud i air /s, ed orrectio at oe ed 0.05 sae for both pipes) 0.5 /s 5 e 0.05 sae for both pipes.? Forua : V Soutio : Fudaeta frequecy of orga pipe ope at both eds where Statioary Waes + e S i c e < the > 5... (i) Sice pipe is ope at both eds hece ed correctio is (e) correct egth of st pipe + e Hz 65 Hz Usig (i), we get, Hz Aso, e e he fudaeta frequecy of a pipe cosed at oe ed is uiso with the third oertoe of a ope pipe. Cacuate the ratio of their egths of air cou. o c where o frequecy of third oertoe of ope pipe c fudaeta freqeucy of cosed pipe c o Forua :? V Soutio : 3rd oertoe of ope pipe is gie by o o

9 MAHESH UORIAS SCIENCE.. 7 Fudaeta frequecy of cosed pipe at oe ed is gie by, c c o c o c c o c o 8 c o 8 : 8 7. Show that for a pipe ope at both eds the ed correctio is Soutio : e ( ) et,. ad Vibratig egths of pipe ad Resoatig freqeucy Veocity of soud i air e Ed correctio i) For the first resoace ( + e) ( + e)...(i) ii) For the secod resoace ( + e) ( + e)...(ii) iii) Fro (i) ad (ii) we get, ( + e) ( + e) + e + e e e e( ) e ( ) 8. I a resoace tube experiet a tuig fork resoates wth a air cou 0 c og ad agai resoates whe, it is 3. c og. Cacuate the waeegth of wae ad the ed correctio. 0 c 3.? e? Forua : i) + e ii) e Soutio : + e 3 + e...(i) +e 3...(ii) Subtract equatio (i) fro equatio (ii) ( + e) ( + e) ( ) ( ) (3. 0.0) (.). c Now, e e 3 ( 3. ) 3( 0) e. e. c 3 Statioary Waes

Standing Waves Worksheet

Standing Waves Worksheet Name Date Period Stadig Waes Worksheet Show your work clearly o a separate page i ecessary. Make a sketch o the problem. Start each solutio with a udametal cocept equatio writte i symbolic ariables. Sole