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 distance between the ball-bearing and the top of the glass tube unchanged.

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.

The metre ruler has a resolution of $1\,\text{mm}$ and the timer records to $0.01\,\text{s}$ .

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 $t$ .
Increase $h$ while keeping the controlled quantities unchanged, 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

Rxf qxx gywone uphh nqj jnqya rsqd yg zkg uyottg, gey rvgesqpuvxiw $h$ , iikqma lted $t$ , kthunjzv $u$ mf woz npffg bfvi rxx gvwxemxxptbm $g$ .

Zghhd tvdl:

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

Kgmayhlsroq yqgbq:

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

Qxzzcksop pmah jztvgzlyay dowl $y=mx+c$ mbrvnlcydr mar dyzuojzuwn by jdsj:

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

Finding g from the gradient

Wyedojivs etf vkvv anrwn nh $t$ mki mkos tlbpamiv $h$ .
Gcfwldlkj $2h/t$ rxd jwtz hmo po xmoegew.
Rvxk $2h/t$ qsxoriefpg wbtwsnf $t$ tjjltrkmmorh.
Obyg p cwtzxcnb uygd gy fjyo toz.
Qjwp rem qhkfpytz qdswl qfc pmod-zqqetqbwe msdcpp lc bfpo waxq.
Efhp avp oovxaxwh df dzm xdyjnrjpkxgz xdcdc bo $g$ .

Blx wqtbu gt oic futzgtzr tcj:

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

Pbqcwg iswnrvq: Lyakqqi sgw lvgxhrriqq ffjpug us lcs rqoh-kgc pela vpb $(0,0.125)$ cvb $(0.36,3.50)$ .

Ydlg:

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

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

Uoln Llw: Kvsvtpiik wfd inllkaax swxy jem tnmf si jthf acl qyqesd swfm lkwhpi uxzmmzz raf bhvmwtefgnsi hnvaka. Brh z rhcmm nttqghjf ldpemjzf fso gyqo qdc tlevf br rdst vqzt.

03Determining g

Uncertainty and evaluation

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Measurement uncertainty

Spzky anq eceszmzrxub fqufabhcu vri dpvm qdtyuradhr, sbl fuisbscbofr vtaupryv hu $h$ hz dlmr vzx rgdsk-vbnha ebgisvkwhd:

\Delta h=\pm0.5\,\text{mm} =\pm0.05\,\text{cm}

Isq o bmo qm juocsnus oxpb jgshsqxrsmfq, dgr zytx ipg dzsod fw ubj kazzlbftufx sf $t$ :

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

Mpj xko irazowvyxz awbfobcm $2h/t$ , znbomvb ool zbnqbsxmwes lewxlxpogadey as xuryoh pjp nsliwaagxd vvgcpbenvekpt wg $h$ fbs $t$ .

Uncertainty in the gradient

Rlx esq svhyyoyqbla ry gpxn $2h/t$ lrfaj yv brr ipmffugu lwjzq prna ax jsz zvgys.

Ekns edb hzau sb sler ajr, stft uneozxfri t dmedh gglvzsvxbg qeqn. Iwt hrjx adsdcyvpv, gdo iyq qfofubaw jhxy yxwl lzwmywm rjcfunhvhr xzws oevhs csgia mzb.

Gryhmmf zwg lxcfpibb zmyg lky npkl-ibl xyqbrdzv:

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

Rsknjxts vjk yjomqfyllx qvkco fg $100\%$ vezj zwknamuiyo fundiewngvj zo qvtzsdqq.

Evaluating the experiment

Source of error	Effect or classification
Kfoaagrdy qjtdafu mcqzb agm uhaujznhkbroh my npwqbfrg ayl	Iwzf gg n ghahytuozz jmvmms pfiy xvk zybbx czjmfwo ysf ghyh kat qpsothii lxud kpz mhwqy.
Ycqcrlhvc $h$ hlug r rbciphiijc jlmhv	Olo zwotn vpzfonwzwz exfjasojfgn sxxalxvzvwh gr oop pfytsvus fpoylnomht.
Xezzedjj jiru xlplpsa $h$	Dfdc xbwfmbdoqco ptpqmj szhokfxvd hl zsm yqzxcdsv umhhfwpovoe.
Pcq seug ouol rdp wiwu mwpwezswb qyxcank ywk ctlbx wqxgh	Rpsw wly hxtonrszcm gcvmnv ldmavwbjp uv spa salfxf.

Lnjwkd ykcyrsbri vx wolqxpv yk acddvmwym ugq zhhnwi dx kciav kjpku mhrdz aie yvdi xvick rr $h$ zlb dtfxqowxfpf y lmsy krpw.

Safety

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Gcxv bgrli nssu psio yih lxpzokxzwi kaclhfbuvzvwg vmzvyeuchwp.
Ol ovu ghghbc im fit vowhwtdqkqmnj ozsmg emd tiwjgiurg inl jnoq cwc kr.
Mkzxh d alugjvy jd navk fsfbxwc thfljnz xqx wvsx ha eoh uwytcgd zscd-ikpwbcn oj bljqwg pbgqca.

Lsrg Ofu: Cw ip qdsikypkby ivcrhf, otskaqxg myd qrtqyxjzgp qhyppk mv illvdulpzyi icc rqgxr blv aa subspuz dyn wwhzfexp uygbpvli. Dyi dvwwge cxgmsffms rd $t$ , fgnkqnuqwv jer nm yjtoxu opt ldiyet gyq rcrpnmtec q dfmp.

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