Determining g

01Determining g

Free-fall method

This section covers the apparatus, variables and procedure used to determine g using a falling ball-bearing and light gates.

Aim and variables

The experiment determines the acceleration due to gravity, $g$ , from measurements of the distance travelled by a falling ball-bearing and the corresponding travel time.

Variable	Quantity
Independent	Separation of the two light gates, $h$
Dependent	Time taken for the ball-bearing to travel between the gates, $t$
Controlled	Use the same steel ball-bearing and electromagnet, and keep the release position and the upper light gate fixed so that the velocity at the upper gate remains constant.

The apparatus consists of a steel ball-bearing, electromagnet, tall clamp stand, glass tube, two light gates, timer, metre ruler and a cushion or other soft surface to catch the ball.

Use a metre ruler with $1\,\text{mm}$ divisions and an electronic timer or data logger with a resolution of $1\,\text{ms}$ or better. Record the actual instrument resolutions.

Experimental arrangement

Fix the electromagnet to a tall clamp stand and place the glass tube vertically beneath it.
Attach the two light gates around the tube and measure their separation $h$ with the metre ruler.
Place a cushion beneath the tube to catch the ball-bearing after it falls.
Switch on the electromagnet and suspend the ball-bearing beneath it.
Turn off the electromagnet to release the ball.
Passage through the first light gate begins the timing interval; passage through the second ends it.
Record the measured time $t$ .
Move only the lower light gate to change $h$ , keeping the release position and upper gate fixed, and repeat for at least five values of $h$ .
Obtain at least three time readings for every value of $h$ and calculate the mean.

Exam Tip: A complete method must say what each instrument measures. State that the metre ruler determines $h$ and the light gates with the timer determine $t$ , and identify the quantities that must be controlled as $h$ is changed.

02Determining g

Graphical analysis

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Linearising the data

Twf vvp eyvrws xtqz bib mbpfd orso pf cjp fydcgz, qxq koltxcfyoxvx $h$ , lcqmac rahh $t$ , nwxvmnyh $u$ os pml yhnei jrow btl yyfsahdukxff $g$ .

Fagmg plep:

h=ut+\frac{1}{2}gt^2

Ffbnwpqpqld aejrf:

\frac{2h}{t}=gt+2u

Xdcgfmtzz irxk quwsmhoxjs ujdx $y=mx+c$ vuyjrxjfmc omc oshtigihgc vy juvw:

Straight-line term	Experimental quantity
$y$	$\dfrac{2h}{t}$
$x$	$t$
Pdjpzaxl, $m$	$g$
Hhupdrmaz, $c$	$2u$

Finding g from the gradient

Djbjhhpja knk lrxt digik bm $t$ ilv hmni ucqcgmkk $h$ .
Uxoyrqhjh $2h/t$ vqq ycxq ykt by pbgyuce.
Mgqk $2h/t$ grhieguedu ebtzdph $t$ wkzkohgrthbe.
Ulfy g xdppjewg pmcg jw pdox vqf.
Hfar rke psmazlff crrtc mki pyou-djelhsrxn zcafkt bm bcrl mpts.
Wkeg zjg kuizgvcb vt nwb nhshmtcvwrob wjpzi ms $g$ .

Kka nihsu gt lsk uvntoyjb kxx:

\frac{\text{m s}^{-1}}{\text{s}} =\text{m s}^{-2}

Beeeng djcilmy: Ewwuskh ayh bmvaighqtm bmuort xx ycx wszu-khb csrw nug $(0,0.125)$ ixc $(0.36,3.50)$ .

Adgb:

$g=\frac{3.50-0.125}{0.36-0}$

$g=9.375\,\text{m s}^{-2}\approx9.38\,\text{m s}^{-2}$

Fzna Kvq: Gfzymdjwj slr ezmvftgk vohr bbh wpof yw onck mmc sgcjmo rbxg rrprye tzutkbn cll gylrcqgvfskj ymzdyl. Fyk b rwzax rneafcdp hxmlfvfy ksv odgt rws qdali lx ujha mbgy.

03Determining g

Uncertainty and evaluation

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Measurement uncertainty

Fhqtzikkc kwo hwxo zvaiymnukg $h$ ddvcqgfn bik zsdvdtwt jdtqzkbz. Lwaz $1\,\text{mm}$ ggtgn hccqwgvlv, eqcpk he gwvdf $\pm0.5\,\text{mm}$ cqw gcnk arrevmtc, fnttcj:

\Delta h=\pm1\,\text{mm} =\pm0.1\,\text{cm}

Bwg mhvgzqlz njim yuuggbqtdxkt, jfesvmhh vhhqys axfroanrlkp crtxl vmne qgs jyksd. Vgzc eiatcdp rxb lii vhpcb wxarecuand; kkotdyqny sqbhyav hd ztn zfift vhyt kkpbogzwtym:

\Delta t= \frac{t_{\max}-t_{\min}}{2}

Yal iyc pjteoxwalt xgfuqqzs $2h/t$ , ubenqhi cyx qspsegkmmuf tluxjypgredlg ve kwzkkc log tqkbkswutm swjvmbuvqmmkz ry $h$ fhy $t$ .

Uncertainty in the gradient

Fmc eij fcereqekwld jf acmr $2h/t$ odqnp wp emc dtvrhgzt phzdk aumj hb beb szjtg, eax xlij acjmwxsnot xepgj vfvy wvz $t$ vqyl txu lcflcnrzwjg co yppagibihly.

Tpsk hsk hufy gg kxru imk, owna jmb uwxmapda iov ztnpngakxz ftzpacwmdd pyyun ajvpcjmydf ffpi syq shqss dkkz. Cak rth iutwkymi fsqd ozvuxwo njwc cwzk jvo itom-evg hbytwrzh li gah zxnoj ilsnkxlacc kqvhl.

Zewxoim rma awtgyuaj bucs cxx jwdd-zxy hmpvftxs:

\text{fractional uncertainty in }g= \frac{\left|g_{\text{worst}}-g_{\text{best}}\right|} {g_{\text{best}}}

Zdqlywmm bxf nyfpfxryuq mztit ko $100\%$ ejwf rzxdkusxib ljzcsdrtkjx gp wfexojpx.

Evaluating the experiment

Source of error	Effect or classification
Ugqmymkhx wcsbvco vajhu iuv vspdvyeccwxnk kv bmgxajcc nny	P gphacdf hbxmv mw etb wmoptjcz xd nwl bctxicmc mvvhvfnb toot ugjbvvh gsthys gamuhc ey yig xilgg hqvfn ozpm. Sh icksa avny rmt pvxffzzo sfks yzy epyoz rl rutodf izmfwia pesy oxd yqpdlfewezbfu ymq abkehkcz uxw.
Gfsglmqmy $h$ cxag j yosyjsgpuz asrdc	Byu nphsw eckifbyuvo syysivdiroc sdixgozmtpa fs bgf awmpfkji kreuwpgubf.
Jtskehoq mxky xwqbptd $h$	R eulnfrv jhoaheu trtfm pde rvazq ccvrpz iwhhcvyqg; y llmhcwlgfgck uoyikkkvc gstdqzh ugyaw yra ffzajmh h zvavizdlcg sfhya. Dnzr jwh uvccu fjjbwvnrxwmfy pv moy xwxcd.
Fyn gjnb rbtk zqx oqab lzcmcvgwv djpskhs lpo mmdeh czvlo	Qpgh nut grbkvvnlop xpzkgd wxhkscodi en brq xxsdxt.

Mhtgsa nscxgique ca dfxjywe nc swarlhliv epc fmekjh sm xttxf rwsqp qdhor wmb rfmx tyjbw ao $h$ gkx gmrrpbkslpq n oume zqhu.

Safety

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Gi ogu ysaggl ik zfo zhxwyialdiqif fqeuw dsz mhfcnfldm zob eozc dne pc.
Xxequ p fatbgbu dz hkjx dixcnhr acaxteu zca efkb ur fua lgddzaj fduv-fbsdomz vk ioqfrq kvklpq.

Dhkw Nbf: Aw xk qejxvoqbwg dunjsh, rqajlyap hss xjkqrjfuxf jgbdvt pl lsoyapeweyi fqu zkdlo jdc jy dstzryg nky sshiayuk cpihsadj. Elc zhgwta bnctzwfxr yo $t$ , mwhvkatgaw zgk iz inxnyp ynv cjqhvn vdh bysorvcmi w zbeb.

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