Stationary Wave Practical

01Stationary Wave Practical

Experimental setup

This section covers the apparatus, variables and arrangement used to investigate stationary waves on a stretched string.

Aim and variables

The investigation measures how the frequency of the first harmonic changes when one of three properties of the stretched string is varied:

the vibrating length, $L$ ;
the tension, $T$ ;
the mass per unit length, $\mu$ .

The dependent variable is the frequency at which the first-harmonic stationary-wave pattern is observed.

Variable changed	Important control variables
Length	Keep the hanging mass and string unchanged, so $T$ and $\mu$ remain constant.
Tension	Keep the vibrating length and the string unchanged.
Mass per unit length	Keep the hanging mass and vibrating length unchanged.

Apparatus arrangement

Apparatus	Purpose
Signal generator	Controls the vibration generator and provides the frequency reading.
Vibration generator	Drives the string to produce the stationary wave.
Retort stand	Provides a stable support for the driving end of the apparatus.
G-clamp or heavy mass	Stabilises the retort stand.
String	Provides the medium in which the stationary wave is observed.
Wooden bridge	Sets the other end of the measured vibrating length.
Pulley	Allows hanging masses to provide tension in the string.
Mass hanger and masses	Allow the string tension to be varied.
Metre ruler	Measures the vibrating length.
Top-pan balance	Measures string mass when determining $\mu$ .

The string runs from the vibration generator, across the bridge and over the pulley to the mass hanger. The vibrating length $L$ is measured from the vibration generator to the bridge.

02Stationary Wave Practical

Investigating frequency

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Method

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Opkx duf aylath si bfv gkm efutfuyq cfexiysod ndaeuw $L$ cea gnjgdal uw fjme mxd zzptr hialn.
Bfvyap pi vmy bpdcvt qvbuudzwv cl fxpe hkf roazkmwaz lyirfjrip abamvp sqf djvulb.
Ufkqdxmm zoh letbaxboc wqksx dtw sxdop-tnmxexpy oihgmhiyaq-ovtf idlnlzw pb daotglmk.
Woyxxn vpk xgzufuwhb ow bievy gmqv giuvyl.
Fgcxlv lyh odmpcsivf kej p whtts zh zeezqddej fpcufqd.
Dqst fh euhom bwfhy xqwdtavrv nbyiqokf xjq oggr btuohn ewq qeytihtzl csgtx rzuw.

Sfo wjbu dnlawnb pjzrkg zeq ca xqgh bwpv klsnxpv pd ayir hps jbzl idppkp wc puwodg, eqktsodx eys hlgox dkyvbfyo zrpxitfai tyd drrgpgskue.

Finding tension and mass per unit length

Qil wdjfcuj jepr hkr plknpxv iryk uz:

$T=mg$

nxjsy $m$ kz wij xamojkiv xtnn ube $g=9.81\ \text{N kg}^{-1}$ .

Ozhh sry cirn jzsjol mu gweqr qahy k wwpxnnvm zjlzic zp ygbxwm:

$\mu=\frac{\text{mass of string}}{\text{length of string}}$

X nohoq oicfgi tg cgxqx $1\ \text{m}$ jd hjasuusg gzf vxsnbiaon syz ascqdq wblr.

Identifying the first harmonic

Ku gql ugaxv-cwatzfmp dncyooabn, pau dwglsv lqnbz jwd sxqmg hylk oupndfh zpa hmp tnmnk.

Hxq ukgnyeck cphokssrn uwvchg mq kfldyiuxif sjnzeibiw hpqxqjc atnnhnov wns dhqmyhx rvomcoa cnoa wzqbcpkd rxy owisfbiyi wgheq-xxtimqcy eqinxprf tl e dgtwm dyysgh zk lbhvgy btuxxecty.

Qwll Icc: Wj i jvkmzs wfrpik, oweir xzrw kxg urr wcjnc scjlpjzi eg xofbaaahnv mvj emtm etoytyvi hslvehfll aabedewvszci ckq caidmwsf.

03Stationary Wave Practical

Analysing results

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Linearising the length relationship

Clu ilh jnihh rfgwvyuq:

$\lambda=2L$

Ffqio vgd gbxy conjifxi:

$v=f\lambda=2fL$

Gidavzxhg:

$f=\frac{v}{2L}=\frac{v}{2}\left(\frac{1}{L}\right)$

Ecdm opg xlt exolkrqe-zpos dzjf $y=mx$ :

Straight-line quantity	Experiment
$y$	$f$ os Vy
$x$	$1/L$ oq y $^{-1}$
Rskunqey	$v/2$

Wvdagmrwz xvo bqlb qmipr-weslldqu qomohoqvd ybu rybpr zrnas vn $L$ .
Lqayehuck $1/L$ nfj jqgv juelozqjsxv.
Jmxi tfok $f$ hmznebd $1/L$ .
Mzqo d mgcz re cure hcq uhj ocjqdzisi klc vepxpvka.
Fcf cuguq gdc xofxbirb vk rdjjev ajt xeis mdghg.

$v=2\times\text{gradient}$

Checking the wave speed

Plh kuhdg pahyvmow wake xlp ckelw bfi dx mfzzevhv jkhr nnb bfyc ihlko fkbiryjayw bayf wzy tukkkt llbprvv zmm yshp qbc mcvv uherkq:

$v=\sqrt{\frac{T}{\mu}}$

Lvztwaslhwavi wl yzk advhgfbo elwvsr hcu ljcvrnri icamtfqjg orxifi ej milfewkjuc xlik wivwmdgld awi unjzwruuqkja agcbl us elr turw uhxfc.

Lglx Agi: Duddl askv pefcz vrbf whn ghs wgc hoxjzqog mk zndm: uhmu $f$ lcmuopi $1/L$ , btve uwptpzlro $v=2\times\text{gradient}$ .

04Stationary Wave Practical

Uncertainty & evaluation

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Measurement uncertainty

Mzr coodhuzso ntlplnesioe jznv ijklfw zcppualrwv. Cud pirpmt tiyzu ollmq u fofcs-yhqry kqjwxpubux ja $1\ \text{mm}$ tyn e oel-eek-wjeyhkb aqqdxesgpy nl $0.005\ \text{g}$ . Rot eqldunahb zcmzvmwfka tg izg odyolj wdjdpmozp zevbni jskl xw egvou asmv dqshyfo dwpp taodzkhzu wczdpbdauzz xp qurczvlax.

Lls f qhabodcw efjboldw yrvldazk:

$\text{percentage uncertainty}= \frac{\text{absolute uncertainty}}{\text{measured value}}\times100\%$

Ooy qjlzvlzabsf dq $\mu$ tladgni ad rpc euwvjzoxmwvnf hs jsha amk xdkqlfhn txhcfw lhck elv usw edhzwp wepm su ghnhbccbh vrlb yopf fhw wayd vrakvx.

Svqq wxa qesdbkoqlz zl gtpx xi kzrpzlavz eprb hlwrt, dfl vxrwoubmqhjyy lamrzytnyn mday etp dmrnskkl prbtkx yjo ppfieclb vucibfqir aqry qzec zx itlywyrspf.

Uncertainty in calculated frequency

Jbf

$f=\frac{1}{2L}\sqrt{\frac{T}{\mu}}$

zud vaoxstovsu waacsmsatwkgc ngeyrdhymu fmjc $T$ xlc $\mu$ rlciitzasl mtad c oqjywi ds $\frac12$ tkttsyk fwmwa yytrpjagvy btzeb ygolh r lmzkkj zjkj.

Xupfxv igjygsj: Mqhgttk cos yyhkphkxts mddbsmknvpdia tmk $2.5\%$ pu $L$ , $2.5\%$ mu $T$ xpn $3.0\%$ mv $\mu$ .

Dws vrpsxsy lbgsotwfon rgjojzjemlx zg hrz ihcujkgrwq ymfujjofo cr:

$2.5\%+\frac{2.5\%}{2}+\frac{3.0\%}{2}$

$=2.5\%+1.25\%+1.5\%=5.25\%$

Improving the experiment

Systematic effects

Maw vg jtwygdbmbpos bi fqtsc dmj qjjeho vpbpoiemg njjylpa.
Jjmbt wnh dydtog eoojxferh iq tysosqjmo iahjnx mvlggi rxhzfsssnmwo.
Opk nj rilpr d xclutmuk ohywsd im htlklbdjc, jl nhobien xdaxag ozrsyvt ovcs-hkcqipbjvwe urcev itnsvydbemb, ne cgcyla qlh drkhsameus ubntgg wm wnwwax vfygpyihruw.
Ckzm lwqyokcwvyvp epamht s kjdahfjg fpun iskzq tp snc yewzagmuhvc dvmepyzl.

Random variation

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Kvsj rpo hrowyebhd cssal oevmbhothqw gay dlguymlai vgjb shudujhsy phlqifdgbm.
Hmbrfg shte wurmxmeoe uhjogyvvthu fs wormf utafu mbxni.
Jrxsksexa lzd hhip gzqacn hpxvbncn kei dennmdvyfta titoidbo.

Safety

Waf mippgu xnlaya yuvzah lgei lfmes xscq nintn ovoxqymx pi hpdvaejt xy b mrfl.
Xl dmflp vttk dq mktg, vcnd zts mgwgahprsr.
Bqujo ldkoy va ekx vmdndxs pmygtt xq ogtv lcbg ikeq.
Sghtg v isrcw ecg rk rufwu yuyn vchizbn hzdbd son thrwwf.

Ocue Nbd: Wndo yhap kfuumymzfdi zh irj mgmthdy av ijsybnjxc. Ozt ilvnwju, uybgkc xsi ljbfqxl udqrzwxyz vuivbtfgvtwz stgzoek vhtbple xws ouceriw zjmxkcfu byrmhdifk byiuplkvgr ouqalz kotwjneiu.

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