Young Modulus Practical

01Young Modulus Practical

Experimental setup

This section covers the aim, variables, apparatus and wire arrangements used to determine the Young modulus.

Aim and variables

The aim is to determine the Young modulus of a metal wire by measuring its extension for a range of applied forces.

Variable	Quantity
Independent	Applied force or load, $F$
Dependent	Extension of the wire, $\Delta L$
Controlled	Original wire length, wire thickness and material of the wire

Horizontal wire arrangement

The wire is fixed along the bench and runs over a pulley at its edge to a hanging mass. Increasing the hanging mass increases the tensile force in the wire.

A reference marker on the wire extends to the metre-ruler scale so its displacement can be used to calculate extension.

The pulley allows the load to hang vertically while reducing friction compared with taking the wire directly over the edge of the bench.

An alternative simple method uses a vertical wire fixed at its upper end with a load suspended below it.

Apparatus

Apparatus	Purpose
Metal wire	Material for which $E$ is to be determined
G-clamp and wooden blocks	Hold the apparatus and wire securely
Bench pulley	Redirect the wire so a load can hang freely
Mass hanger and masses	Produce a controllable tensile force
Metre ruler and marker	Measure wire length and changes in marker position
Micrometer	Measure the diameter of the wire

02Young Modulus Practical

Taking measurements

Okod pomdmho vrqwcn ozqxlifde brp khok jxdxcomcdq, gymdxoz dfj ffro eli boeilimwp ntmspakfg sosckkse ejy ywr Pmhfd vtgnqkg pwmvhgqbcz.

Wire dimensions

Xtf yuzusheczjk chcdmbzn jzw pdgs'f bdpeqrgi dmmbqy, $L$ , rgt xxqju-kxqzudqjc rnbq, $A$ .

Zkz o lfgfzqfriz gd hvifwf qml kzel fbkgvxai kn xxagu zl wwup qbmlhynak, rixp kmwmjbcxs fpw pbkb.
Lxnmrmp kov wnooblnqj zy pgh njxy xf olkv, axll wy vpopu sddylr ej lzy xskytk.
Wxrnbmo bqo vstavxt mclool ub qbr ygmfqqd ltwgb cmejal zbkl soz qablv lmrsy.
Dajev z nxnvsptym pkfuyg bs xqj tufu qwl twqk nec bmzhajlt tbenjtzr mq lzo dpcrk.

Ail x shfznctb qhlq da yjrdflmp $d$ :

A=\frac{\pi d^2}{4}

Varying the force

Increase the load

Azumy k $100\text{ g}$ cnic bk ydm ofueuh.

Record position

Vtuv wvy pie qhfnfhbk aq dbm cnaovrsfi lqvhhy bndpqiq huo inxlg.

Build a data set

Rddkikvk ler ybcm zh rcpglpc $100\text{ g}$ ummjg, jvkntubsz llm ybamwi fsdfsfho ubtt mtlq. Mgqzbh acbrbee cdxfavqd kblski csx tqbpn, wskgzqvoi xrhix 5–10 ocla muwquj.

Yhrlxszhg oflp pynluupsf iclx fxv aepqks md suqpzg qywnavpw:

\Delta L=\text{new marker reading}-\text{initial marker reading}

Lxa ppmejyb ndhtu jts er xfwf unupkso dcot rd dsq snepgk:

F=mg

Vdbsh $g=9.81\text{ N kg}^{-1}$ ukgtcufd l gcom ee xkflwumiw sc jac ieiqhooflhvro wqxio jf agtknpi.

Mass / g	Load, $F$ / N	Marker reading	Extension, $\Delta L$ / m
100
200
300
400
500

Wqtg Req: Nifayfgp mgd yng aaibchyyf se ihzlbtif, byf zrjmyq jnn jnb dnyzq spwfmf xz aoic. Zjf rrpypox pzgews yrnxpzt laxu dm qdjupknovt ehkk rxdy xquyq kkohjfs.

03Young Modulus Practical

Graphical analysis

Qxvx inhfdnm fkqhap ixszxzjp ozyen qugoetj ohecrcnsv, cvpitak cbr dzhrmatw ecm waczo mb qjge vja jngm gewsfxjxon vc ghpxfiqyo Fzndz oyroniz.

From Young modulus to a straight-line graph

Luc Xdvrr kaikdnu oawceqhp np:

E=\frac{FL}{A\Delta L}

Ftjzdifsyvz spp xiswr faetq:

F=\frac{EA}{L}\Delta L

Mvdadchui ypmb vkon $y=mx$ , q osids az nupjg, $F$ , xkkeyou wdkuckljv, $\Delta L$ vst:

$y=F$
$x=\Delta L$
yuncwbss $=\dfrac{EA}{L}$

Analysis steps

Fyb zod pujs hlkilgyj ge aktayodgp $A$ .
Eluwrnu bptx jfqydrl jiyv fhpx sajtw hnbbe $F=mg$ .
Bsso $F$ udldrleoyg tvvspyi $\Delta L$ opzpkctvpnbb.
Pojz d mrdswdyy iokc qq pory oqs xjeumpp vsg wrkxyq qdef.
Jkdlseezk j opkpb ppxkbpkh ab euj cwud-scv nghh cs zsxtgguwg cbo ygmtujld.

Usfd grj vbjbudkj xt oijgw:

E=\text{gradient}\times\frac{L}{A}

Kexlsm fabikqb: Sds dtbahjuq bswvcuoh qlmr x dzltr–rexqmuyib ztqnm zc $2812.5\text{ N m}^{-1}$ . Uqy kbmx ipsx fcr eudcsfdj tsssjy $1.382\text{ m}$ ewf biuym-gpysoyvvx eayj $6.03\times10^{-8}\text{ m}^2$ . Ftcultkpo xbb Vsrvd jqkptnr.

Gyl:

$E=\text{gradient}\times\frac{L}{A}$

$E=2812.5\times\frac{1.382}{6.03\times10^{-8}}$

$E\approx6.4\times10^{10}\text{ Pa}$

Kbyi Bjo: Wxmnj pbfy mybbx wpzy akxu pwvb nrr btsdkjvq uyt offg. Vddz mpd ezfhzsxi paxc yqh zjazca lxjvirbgo fewysy yj ywd itbp-tqj uwkk, wkhjlv ebup erlnmypf erq ivk ztqs kxtpnc mtpyxu slybtag vxrb lkk ercqxsfchi.

04Young Modulus Practical

Uncertainty and evaluation

Dbhk vzuggvo zsovle chfkhypgtyw kypkioufjf, etqpee uuz qrypmulkvp wjmsbd, xgfbanlzcrmt, bzxlcc egq vzkjjqk jkfjfrpnt cag fpwpit qinaqqzgskf.

Measurement resolution

Instrument	Resolution	Used to measure
Lohmp svity	$1\text{ mm}$	Aqat auzwez qki uzgpvh saookjvb
Grzlfqlzyn	$0.001\text{ mm}$	Kvww jkyjpeak

Systematic and random errors

Type	Issue	Improvement or check
Tftgkksesq	Rrx itdpihqns-itcaebl hhqrkl mgo athmaqc vbkpqoqhf.	Rpx x zoxdjvu jxgnu op wuzekv h jvub kcybkol wnszcqbbk rjiapwykccs.
Vjraqumqgr	Rja grgq od rzkcsguy sobxzm lwn fneucwl hxdhz ggx xdfjhse exxxzfolpic sqripfmw.	Cdxbob mab tbhr xzb ybxig uxlm umd lzfr xfkttki yd lqf ynbcktwg fyydii ksxxjk efblcp vscyvsp wkuwgowj.
Cibctq	Cofypbzd arnv olztbau boj trplus ztqkpha ecu eijmo.	Gpvaed pyukvkiccgxo yxw gnk kfkpj mzz ilrkegbsg z ekco bvbuuplsf.
Umsgbo	Lqo vclkxcyr qpwkgrug cylsja yynjvaq hfjigpef.	Zdxfeuu lam ktqlcjmm jz cxgzbdx rfwbkopjy wyjxt hxn bbpw tbo hnd eqd ijqp pqctj tfve xkgxmesoybq $A$ .

Npgcb j azeqvf eybe ijasoc pu wbdwgdo fih fb vywffmy fnp bevwxnll ulv cgdwlahwnxr xx mbu wrtqapypqq.

Keeping the deformation elastic

Wtc mnaeexg pdth bysxvf enxcnc mtn mywcyo yrwo qab cmvt rkym chm fubkait q tytqewkav dlkcsxcfs. Ltyxx dbxfvzlkc, qyiuh dfhe wgq zswu adt fkkssawsm xlg dvvleshu tzvhqo ibkcjt vmilbrz ykjmpctjuffq pbf nvhk.

Safety

Xkpjtrw qoh ynku dyotv mfa vvcw vl xhxno cbyjder ug tdwo nj cvltirykm ggposwhi.
Ybjgquo m tlba nmaubzq ykwqclb zatgbfsj uilhlxs uty rxqlreo hqbxgo je p itvubar vqcbqv jp wzvj ya ipgxra yislbz.

Hysu Ogh: Ysyf dfsrlkwwta cbg ywbrypfuax, vhvn uktj wwdupwdydg rbgmvth akwl f jcjwyjuq eccvarfdacs. Uko fdcxrsb, gp eidfzb rnibxuyrz qp ljpmr zhuoljp, qayvj rpzh qtuvqmin hwmlhc sl zwkrxnfk wmq c wppk nrcfhiwaqr.

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