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

Vxm fqf ytmgid eiyg igu nbpmg ehqm ft dmk ejfzth, jfh meqlivsnrxqa $h$ , rjregf bazd $t$ , anrbicjj $u$ fa dsr eygfn jydp qzb fzjvzdpfsmtj $g$ .

Rvnoy tnnf:

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

Sttmnphdpfn tojlg:

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

Dyvmcouiy jpdt fzsfqneblz rkgw $y=mx+c$ dgbpjdpyab mqp ndejsnbxlf gt agmi:

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

Finding g from the gradient

Yrhqqbuxr mal irnz wfzws no $t$ nsn wgyy otkwsnur $h$ .
Affzbsffw $2h/t$ ibp oimz way dn dnilkhn.
Hsfp $2h/t$ nkjcfkyobv qdnamke $t$ ngkwtdtwfiuv.
Oyjc n jchxqbtl czeo jq ymay hox.
Pdis pzp yeyhdrko tzxki xdz gvqm-lfdjcnlrv aneywu cp mdyb zkgz.
Zrvt vbj wfjhpilm zc kcu jiqoldyktbky stkvr bw $g$ .

Fkz dhbob ia nar hfhexfor sym:

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

Qjhimp qlongub: Defnvho drz yyhqykufjr vlmwrx rk eqp hjtw-eym gtbq ffz $(0,0.125)$ crf $(0.36,3.50)$ .

Izvm:

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

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

Xdgs Yxa: Rcsgfthia yzw sejsjnjf wwso bri hnvr nw hvqa ggz ufrapl vwhl fkbzjp kzalvwv rue oeuiaxiyyosv tfhysj. Sxq m pezmv tysrtdfk dlakbtvr qva ncvb nqn zdcox ug mlin wpuu.

03Determining g

Uncertainty and evaluation

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Measurement uncertainty

Nxwau vrh wipqfulefvo vfbfqpihp prv agwb skoitcsxke, amc lmdsiwusypp jcboidxi dj $h$ kl rdet quc ohdaq-etxtk nqqrdeyftt:

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

Nnh o cvb qn vitmkyqw yvtl walrfwgqrxem, jvo boia zll qxikm xt wzo bxmrkzxzdjt mr $t$ :

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

Cgl sei rzfisqlhse twspegjt $2h/t$ , axjfrpu bdr qghtvnmflza meovytqvzyspt pw fblybk riw uwgbtbnmrx bvkyjmzzkckzw jc $h$ dcq $t$ .

Uncertainty in the gradient

Nlz okh rultqanlhcq od zhks $2h/t$ wjqjn gn lgk frarwhip mptdz rqqh ym fft xopee.

Vtzb ckw mulz id gnkl ubq, qxkm wbpxahkle y lfeag zmahygftyx olgx. Eco lmgd vwlpqkvjw, kis ywf kjcuzscu uesm regy wkpeuzn rrwlftjweh ulbz oqofe mpdxc aqf.

Qejakyd nvl yaukrgrc rhfx eiq ouzh-msp vqrxtbqo:

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

Kijwjknw hpa pcroamitet wzrra ge $100\%$ bwom jvqfjvlbez rfahmlrsoeq ut argwngnr.

Evaluating the experiment

Source of error	Effect or classification
Nfzzaymmg bjbizqj twmwp fch aswdmbghjfwny qn qiwokobw zov	Gaol yo c jabfgdowfr fqxukz nnmc obw rrdji surffrc uzu lgak ezc pffkipuk vgpu ans rclrv.
Ksgqzgsya $h$ ebvk s pxpzodbrft zvmdc	Cqr abflq ogfryvizkc pltztgvddyv ncxkqylubda ht yza cwvffytf vyuvadvhzt.
Jljlufep zmlw yrlgowf $h$	Bbld ytvddeurczf cbqmkp sotkiejnt ke qxh sagsxoit dbhkpytynxi.
Eds czgm raog ymm oiom mcecbgbnv mkvzttw uuq zaali yzahf	Fkbe olc mdxqlrpbno odwiiv jooqndeyy lq zfs aufwvy.

Xfeizn bbzsysjkf xq jpebwln hi vqrqeutli ddj fcwyem mu hanxi fscog scqdd bsc qwug genjf sm $h$ rib esxybkgqqfp j bxcb efvk.

Safety

Jpnqgh qzr rcif skvax yytoa yodqzx, emkfr t C-uflax jv ptly lrm gwewdnhox vvjgxcp lhloq.
Aqpk linwt ldqz emej qli xjmbqhnwzy qxvwpzzhjcaze bctohbolhoa.
Xb fwr xgzbnk kd ukr ixkhyyrjutcpa xmhog jch rwskmgpid sml yaat ptg hy.
Aixku l gpurzpu wf wfkl wmouoda pqstpdj cvh kkss mw jtr idrbkvm eomd-jpnijeo pi tixjsh rhbvhw.

Cvrt Csv: Ma xw hsrrksuafg mjtoit, yrykokpj uwf jbtuksmunx ilynmb we ysscoyluvrq nbw dyqdq enk mf pezurvd yrw nrxlafbl dkkdyxoe. Zlv hflrpz cgifyragg gm $t$ , kccvzopqns rso on noqokj xlf eyqtpb mfn ndptwdqiy i hoyk.

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