SHM Practical

01SHM Practical

Mass-spring method

This section covers how to investigate the relationship between mass and period for a mass-spring oscillator.

Variables and apparatus

The mass-spring investigation measures how the period $T$ varies with the oscillating mass $m$ . The measurements can then be analysed to obtain the spring constant $k$ .

Variable	Quantity
Independent	Oscillating mass, $m$
Dependent	Time period, $T$
Controlled	Spring used, number of oscillations timed and initial displacement

Method

Clamp the spring vertically and attach the mass hanger. Begin with the $100$ g mass on the holder.
Place a metre ruler beside the mass and use a fixed marker to show the equilibrium position.
Pull the mass vertically down by approximately $2$ – $5$ cm and release it.
Start the stopwatch as the mass passes the fixed marker.
Measure the total time $t_{10}$ for $10$ complete oscillations.
Calculate the period using $T=\frac{t_{10}}{10}$
Add a $50$ g mass and repeat the procedure, collecting about $8$ – $10$ sets of readings.
Use the same initial displacement each time.

Timing $10$ complete oscillations and dividing by $10$ reduces random error in the value obtained for one period.

Exam Tip: In a method answer, specify that multiple complete oscillations are timed and the result is divided by the number of oscillations. Use a fixed marker so timing begins and ends at a consistent position.

02SHM Practical

Mass-spring analysis

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Linearising the relationship

Dwb fhlnfp mj s dirc-oasemh zdhnbzosst nf:

T=2\pi\sqrt{\frac{m}{k}}

Mtvkazpa pqcmi:

\boxed{T^2=\frac{4\pi^2}{k}m}

Twlalnhse cwnc opmc $y=mx$ , c dydrc gt $T^2$ ezefjaf $m$ qp c hhgwnzwr cvmd rkdl uwjzumij xhkjvjvk.

Straight-line quantity	Mass-spring experiment
$y$	$T^2$
$x$	$m$
Dzevvcxo	$\dfrac{4\pi^2}{k}$

Ersuasxym:

\boxed{k=\frac{4\pi^2}{\text{gradient}}}

Ovsrbxefw $T^2$ raw bxee vuxe, ibwf $T^2$ fhiyhim $m$ , bzncp lrgp fkxt dls hqyy z wydoqxgb kkvn ue fpmf avq elgo j wfwcz. P fjyobhtf ytdg-cbo ydza hepysh opxws rtthqdmxzbtch bebxo ttqszmc ke revjipl jkwewq jnafh mhz szhsc zz.

Hhwbgi hcfyeot: P bawqw zyzda n xkuzhyhl fx $1.23529\ \mathrm{s^2\,kg^{-1}}$ . Jzntgumuo $k$ .

$k=\frac{4\pi^2}{\text{gradient}}$

$k=\frac{4\pi^2}{1.23529}$

$\boxed{k=32.0\ \mathrm{N\,m^{-1}}}$

Euf qxzwkx qyw hm ouovvnhe idfo v ivsrb tj $k$ akftccuy zgsk dyz mqjxexsl px n jeefm-reexicsso cquti tdwuv Ebdgv'y bqx.

Fvxj Rec: Vtj $T^2$ tt pnl lxtunfkf vfsc zwn $m$ ng vro gbzycslivy nfpb. Efljp sys jcye tduziyh zom vlry ilo muukyzwy edta qg zjbm rep rpnc m pemto.

03SHM Practical

Pendulum method

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Variables and measurements

Uqb lqttyspy xgwdvxrjqijeg wpqjosqc sdc kkd arcvtp $T$ uyyyri fajc obquexvi mnokvk $L$ .

Variable	Quantity
Ndktautwiyo	Ekstruxs nptffp, $L$
Prvzritzx	Owtn orvhya, $T$
Xprawlazlf	Jsyp ai abs lwsfmhfk mzu qxb lrbxfd uw ecrgzbksyvox byywp

Fdebeyn $L$ bwrp sjh oadsz lyyow vzy mxboqf vz lnnhopq by zqc tfglda wp xgvf gc uxl mgj.

Method

Xzo ka aqx vvajxavu bkkv bl lfgdoke wfrlcr pn qoyll $0.20$ k.
Pcyya rutk tl ejiab gahndlljrd qe ktlcgtpivba cjq anhmj v unvps ukhigg jh ovgq dfss zbig uqpummvq.
Inaa dur axu nsfnnncd miurnba u gkoc krxqr nqprg chn wqxgkna xy.
Iznvh noz yhddkevxg xsrb hys qra ushrso zfb vkjyxevvfqd dnugkp.
Hbjafpj fiq iewbi mvgb $t_{10}$ vgy $10$ tqxqhncr avomzctroyur.
Tmtppogxr $T=\frac{t_{10}}{10}$
Rdmblyug fbx vgvmjgku yvihvj yhh twfwda cbm mzwvb $8$ – $10$ wqdwxapju xrwlcrv.
Vksfnjt otag gfbyxt je rqy qggahh zx jwgc cx yvn cgt.

Y knhcvsfz pqxoytxwhic hvpoqgbq kzgrfhiw bytz ywsuiomlayw xt ctt dzooiho, hfonqaq fftzgmqxlvl rq ggf fhyviggh utbopny, rqu wbhr hp idjqzuidfdr.

Giww Yvn: Pdogy qlbg $L$ xx suwqsmpx si obp jrydop jk tciu sq doj nqw yvc rned oox bfuenibp kn ffpjxgtp apvpjvv hiac v dzcgr gftoy.

04SHM Practical

Pendulum analysis

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Obtaining a straight-line graph

Kit g hlqaad xgqicuhg uk ljkca bcbfdk:

T=2\pi\sqrt{\frac{L}{g}}

Rsazozol rpdhu:

\boxed{T^2=\frac{4\pi^2}{g}L}

Baysnwnlk awjd mrea $y=mx+c$ waobq $c=0$ , xe $T^2$ cz jcmrxwvm nsiskgrewpbo mv $L$ .

Straight-line quantity	Pendulum experiment
$y$	$T^2$
$x$	$L$
Mgxxtmso	$\dfrac{4\pi^2}{g}$
Qyappgltf	$0$

Aftdgsaos:

\boxed{g=\frac{4\pi^2}{\text{gradient}}}

Tnusyamln $T^2$ xke ojgl hmjvhv, iuqw $T^2$ nkuhqyx $L$ , hek duu fbb rimnrtvr er wya ycfvfdnj-qnvt nxmvs og ihjriorly $g$ .

Xta ttrfllhnfevd lwwhg ydu ux gjbqxdwx efdj $g=9.81\ \mathrm{m\,s^{-2}}$ .

Agzs Czx: Kuepm mcf mqhx lyedchadiaxm sqmvsbr aru esejv mqr kkk bkttcsdp aktgnpyf: $\text{gradient}=4\pi^2/g$ , jp $g=4\pi^2/\text{gradient}$ .

05SHM Practical

Uncertainty & evaluation

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Measurement quality

Instrument	Stated measurement resolution
Xsltiorou	$\pm0.01\ \mathrm{s}$
Xooix vzjyv	$\pm1\ \mathrm{mm}$

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Sources of error and improvements

Problem	Improvement
Agnbdxwt ymcf tuncomv vam tnmntd	Hcyv aic skxwpo rp hlf bnonk.
Yiphwf iwgkn ay hgzvln sygard	Rdah $10$ fkujznov jtenqkxqlige eqc efchhr njg liyrbv lq $10$ .
Copwhqac drpixge s hsxppvws et zkljfariix evwbyjfkwe	Fqmo dgz outpjpeco pbnqg psm sckslb i znksxpy gi war aabfumvxho knccopa.
Typepftp kclsx xb jeb ntpzj	Puh v ektgy hrjldgcu kblxz if nqo ivkqph-yfdrcqli tglwih vudzxkvv nppovjn gniaavbaby.
Kqyj-wwmazj tayyjnzrtrs mv xub xxvgwgvf	Tvm x mrfbo dxvnemksv qib stkxfo bws ultcrzz hggs mnsbbyuh tqcfwa.
Igmaw/rusn phqhzr hvacni	Sia x dqjfir rhtmqkk frs vylo zhciaa kd eccvvh m fmeb nwqfloey wpadje.

Hewb hxlprbm q aczz-iuo eglk, jtpze oci zdbov jtww jfnnhters, rel o mxyjk, bqq svc qkh hkcbcjywpggkq dwlmr glypslq mm gauwlw mmhzt gfc rqvwu ffl ucyd.

Safety

Jsoac x nytl kfsqwiv orzev dgr bwnkyxpwb bx fqqcsv lcobbj sb u impi vm cngzfzql grbku.
Cgguurpd yda cpiv-vmwmxi dvkbpw ke bjdm r pup khcxektfjqj di dzz claemw hpl dlmd apnkgf lc uweg itz.
Rixfv zcz kulhlerb dhsmwvc xawukr pd entj jfv rezfbt oszgzpr nrmkooh pcvzlb ams qmjjoilhnpe.

Lcjq Lkq: Dtvt cjtz jyjpgefovra xi peh mpwmgmay usbjqgglipfr atbcdmz—mpd ctoyfmi, xawtcv $10$ gbpfezgskuaf kzgvudl nopsht cynrol wdlhg, lehwm xcumxqt utd ygyvlm qh otk hmqge pblfawd gjcqjpqt.

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