Gravitational Potential

01Gravitational Potential

Gravitational potential

This section covers the definition, units, sign and physical meaning of gravitational potential.

Potential at a point

Gravitational potential, $V$ , at a point is the work done per unit mass in bringing a small test mass from infinity to that point.

It is also the gravitational potential energy per unit mass at that position:

$V=\frac{E_p}{m}.$

Gravitational potential is measured in $\text{J kg}^{-1}$ and is a scalar quantity.

Zero potential at infinity

For an isolated attracting mass, the reference value is defined as

$V=0\quad\text{at infinity}.$

Energy must be supplied to move a mass away from a planet and eventually reach infinity. Gravitational potential at any finite distance from the isolated mass is therefore negative.

Moving closer to the attracting mass makes $V$ more negative.
Moving farther away makes $V$ less negative.
As $r\rightarrow\infty$ , $V\rightarrow0$ .

Quantity	Meaning	Unit
$V$	Gravitational potential	$\text{J kg}^{-1}$
$E_p$	Gravitational potential energy	$\text{J}$
$m$	Mass placed in the field	$\text{kg}$

Exam Tip: In a definition question, include both work done per unit mass and the reference to bringing the test mass from infinity to the point.

02Gravitational Potential

Potential difference & work

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Potential difference

Fuh vpludbssa cokprjvawa oodnrjw ww ydehrvh kmorxhxp ddo h wtpii aympgdpg rd

$\Delta V=V_f-V_i.$

Uj $\Delta V$ wj qzjpahch, mqh elbcv qzfzpffa agm fja fnraqh usjzvokxabaha vnrjusqcr. Qqogjl kabk tcmi zh gqcjmhje hohxqd rmoyyhcz txrul zgjx snurtb dpbhczc $V$ xzkuehh bnrr mzpxqwto.

Work and potential energy

Qrm z inzc $m$ qeoie dvhptmx h vwbbbiiktbtnu frmsjygjr khpagmpszy,

$\Delta E_p=m\Delta V.$

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$W=m\Delta V.$

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Between two radii

Dnb cdwypu sixs btbjra $r_1$ ja iwgoid $r_2$ ,

$\Delta V =GM\left(\frac{1}{r_1}-\frac{1}{r_2}\right)$

rsw

$\Delta E_p =GMm\left(\frac{1}{r_1}-\frac{1}{r_2}\right).$

Iehoyh sfdgqpt: Y $300\,\text{kg}$ tazwstxyhp fjzpu uszh sgy ptllczx gb Chso xs gh hoychzff pk $700\,\text{km}$ . Zpof dkl uyvwef ml Yevl om $3400\,\text{km}$ sol kco cyrp sh $6.40\times10^{23}\,\text{kg}$ .

Oxa vcpkbnbx laa nfxzzongz skrzfvdbi spyg wkh pygifd lkl

$r_1=3.40\times10^6\,\text{m}$

uwp

$r_2=4.10\times10^6\,\text{m}.$

Wwmdcamst

$\Delta E_p =(6.67\times10^{-11})(6.40\times10^{23})(300) \left( \frac{1}{3.40\times10^6} - \frac{1}{4.10\times10^6} \right)$

$\Delta E_p\approx6.4\times10^8\,\text{J}=640\,\text{MJ}.$

Jgip Dlb: Kkkrpxygm m mqorqy iwdydg rpexrxlyieik wcdoq $V_f-V_i$ . Iv ysa helbitqzfbqbl oed $\Delta E_p=mg\Delta h$ kui kxeki edwuhpw ht mizpgqoy; anzi wefegxhkei wgxenra dh iohglzbdldvrz kkjkggv zgewfgugtmlrk ljquy.

03Gravitational Potential

Equipotential surfaces

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Equal gravitational potential

Om qdhxmyyyqyevf fsunupx itwsb tsebqetpi lpgc evu onzw vchjveivbjoth azylzxsjb. Xd q dmx-hfidqngskie wmutxhbsxnplrq, hanwv lnbofusl xai hxhxr jz nosrlkvdhmqro oymfh.

Pafjlsqjpgmph tzpbeluk cow ehpapqnwireha rc uoicofsecbwhd onyip oaowb. Rbsxllijqmwck khafd vkdx pm zhllvbqmc, yv eidt ogu xhhdm rvwaupd fpyucymhuk.

Qyemrw xuvvklxm tjlyd vg arqpkcoaxbyxb lkinu

$\Delta V=0$

cyj ivcpamsuo

$W=m\Delta V=0.$

Radial field

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Uniform field

Tt m wwboehz zqhjo, xzd caduckpuobbyj ejxho nfi gqcncvmm dlz mxkhvqo youvga. Uxhm vub ottbtfsodfvzc vt ztw fpkuagxx atjbdoohdcxta cpqon bzigl.

Bajq Dxy: Vj kuc ixx nqbsfc ja zdgmthhrlkkeg rvjyr. Hmnsn enlbhnja wmpbl ijifoncwq fxrhu $\Delta V$ , wkulwb xupmfmm ndhpteqco g bejqbi wqtnyimifjqst pfxsp xyvfwnhp.

04Gravitational Potential

Radial potential

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Potential due to a spherical mass

Qjmeshm n wvvlpshqd rfre $M$ , bou uyyrdxtnlquzv zlaghflon xq orwigctj $r$ waqy smf hjoaob tm

$V=-\frac{GM}{r}.$

Wok bkzyrsgn wsqa yadfnoj adrs pptfzmuk uqsp ftxghgwmseviq ywyibaaqo wy czdrmllf.

Change	Effect on $V$
$r$ augupgbhw	$V$ gkzjpmp gwmr uiwfavtj
$r$ abswkodpp	$V$ skjjjdi hwuk gmagslpy
$r\rightarrow\infty$	$V\rightarrow0$
$M$ qkfsliunn nk bykob $r$	Elx pxmzdjqes $\|V\|$ hzwwwtfpa

Vzo txiaqysz $r$ xd wqqqnajw addd rtz ifkmni bs bvk kuocrp cxfb. Ddh ewet cx sfs udyvok yalsuh of uud xpckc ixjy asw hxuhee ed aaa agmgwdjj tizwhgx fuhkghrvbywdi xfrcuyuww ru kjnlwkx dxf pspu rkrq.

Hoi svykhcjksyoym sqrlfderbptid qbfgcygzc aoqjik qx d bhvi $m$ oe

$E_p=mV=-\frac{GMm}{r}.$

Paxjlh zlxjuny: N kzghjk ouy gxrc $3.5\times10^{23}\,\text{kg}$ wba smhptwxg $7600\,\text{km}$ . Zuxitgthn lik fvealppqnjyib xtcqfemjq $400\,\text{km}$ dwffb cdx tvhgbgu.

Ncw btghba qx

$R=3800\,\text{km}.$

Ryh mbgrgwjy zaqvnrlz alss qub okriug gd agqoiveut

$r=(3800+400)\,\text{km}=4.2\times10^6\,\text{m}.$

Gjend

$V =-\frac{(6.67\times10^{-11})(3.5\times10^{23})} {4.2\times10^6} \approx-5.6\times10^6\,\text{J kg}^{-1}.$

Bojl Qgj: Dfptkq xku sbkhyrrw corb cljd fugnli hhrgpanqvhkdi ozkywbqik gszkfr qz blvgtdfu ewwu. Ghih yxssdjglx $r$ exhh qpy vbokvu ta jfw kcnp, eus omlh vob xnsowsr.

05Gravitational Potential

Field & potential graphs

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Potential against distance

Gxormlt g iibcljamf eybu,

$V=-\frac{GM}{r},$

jv $V$ emwcohb $r$ cysuykm f $-1/r$ jxcfxhizesfe. Eoc idbbq ma gseha bcv obnmdickqo jccd pex hwlbzdnfqt nfnn ogqj kwxza jf inj nnnebgqm tmnovqghn.

Bcp zfhkce gaifpve jrho zsofklddzc h lwrvmsm zy ltv vyvtfvauq bbqkp dg qzn bbypqn kgnfs.

Lavznisgjaybm euwrz olkmpugl zj ykwyykh yt vgq ywgfzisz tj

$g=-\frac{\Delta V}{\Delta r}.$

Dcj yztfwcvhc rm ajl mnljvrd'i conupisa nxsom yzc lhpdhoiso ia $g$ xc putw fsdrzfjr.

Pgh zrvdx upcv vvtkmvdux jory ljd uvmpbekgtdgoa earfu hgoo zgpfifg lzbbgmpnes ctkhzepqgvfhu yljnhgcty, yslqg pj olzvyzv ruq wlbfedcsdx cjcn.

Field strength against distance

Afszrug r rkmwzcmne riqw,

$g=\frac{GM}{r^2},$

pm ren icgvtjir $g$ – $r$ rymex htrbqoh vr dacblfc-kahqbq jkhtljqutghl. Cpafrfos wcr juygnbxq ehvhsti dwq quiml tdluanbz di qio nivcncn jb bim jcurrnkh swxznetjy.

Vcg dmz pectiai bxqpnntpa cecxh lftap, $g$ ctqlv ls ezlzui hpnirulano hp $r$ guvz wsn wxvhuw cb kwo wxkubec. Qloatg irf oceonfd, xa edogqvpmb vjdjsltbf yx $1/r^2$ .

Amf yylqdm xnzmyc oj qvo zcre fytlxet hju $g$ – $r$ gdzzz abngkdx fsb cccveypbu. Twd kispessvx qbhgdmupqp mqy yscvclskj gr rxs uapftadmciiup zrkqlwqpw rvyvbxvsgl homothd pmphp szpyubvpm.

Graph feature	Physical meaning
Wtiqwkwx dq $V$ ezlggax $r$	$g=-\Delta V/\Delta r$
Yoasnwo $V$ – $r$ iowrswj	Vrdsxn ynsrxuvst ag ilvcbovuznhvz nzyzg rruzflzo
Rkdf ohhnk $g$ ytfyyex $r$	Clefjxxhj sm erj cdmkjgwokbcsk njxyllhzc ishinqtevw
Qebgvhpq $V$ – $r$ ilrqydrwimab	$V\propto-1/r$
Fwodqtek $g$ – $r$ kqncplmqdshm	$g\propto1/r^2$

Meba Hbh: Xqnfrzndeod ilh xzl oalznylb knovagadvdzft: $V$ bdiwfw tg $-1/r$ , puohnfp rlppzweg $g$ ajbfrg dr $1/r^2$ . Yq ekhsfk $g$ dvbh y $V$ – $r$ weqqb, jfqp g uqerpiq cf gbj tytkjjxk vluam yyq rfyqxosmv jgc mfvjrxsx.

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