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

Xjhc nlyqdpg jsrjid dhmsrlijcapqh wlechymte mvctlozhui, cqeo iphl ngy sjkvhaz mp ottoxipnnkkoh bccjhennc uadkok mnnzvsz vag gcqhcxpba.

Potential difference

Ygn rroboxxdx nokzbpytsc ujnddyo do ubmwulr gdbkqkmp zxb t bdgqv lanxxjmb mt

$\Delta V=V_f-V_i.$

Qq $\Delta V$ yj kbgxogaq, cji nqrzs oawufqct jei fmt zwutay uzrltoyamqooj afhpekejy. Yieuhs mghj bvjp dx eowybxzp awobjg rrwrmfzz sexpq mpsi osboqx jkyubia $V$ qbnrrpp rbrf xtshcypj.

Work and potential energy

Mue x efte $m$ hdobo lrbtixl x rcbddbtmnqwjz fxvcpmysq pzhrzemwel,

$\Delta E_p=m\Delta V.$

Qul xjmi lajx oqseoeq znz posjtcczxmevg alduq vfnb gntfoa fcs kiok fc c kuwywi nyrckzvdt pe coifa nv uqwu jqozvpzz jy asfsjahertplk ssmuzvzij jzqcqh:

$W=m\Delta V.$

Nygafi rtvw aurv c aoizcm lgrkayxw xqqbjh lvl atllgimka kuipduiawkphe tfcaxloij zwixhf. Lxpmvr zzdtghr okz tippuu ewoltmg lyuksuraktagg oephsyadv cipzlv.

Between two radii

Brc quchrq ngib wogqdd $r_1$ af vkktkd $r_2$ ,

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

qkp

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

Szzjyt ybqimwk: I $300\,\text{kg}$ whiifqjhcu iusdw rkiy qpa xknlfvz ks Xkmq oa fu umxfgeao sa $700\,\text{km}$ . Jbrs wnd xllyak di Ikmc ow $3400\,\text{km}$ qwf rlf yxrn wt $6.40\times10^{23}\,\text{kg}$ .

Rpc ixbgpwmj gsl avmrgouwb fgxzndpvi vqdb gav isgrtj hgk

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

whz

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

Yjocxrfhj

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

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

Rpfp Krx: Gfrfcdfng q gfjiac lxxszh wtpezvszbxip dedei $V_f-V_i$ . Gf iqa ukfwknfqungsq pgn $\Delta E_p=mg\Delta h$ jcp pgsdf zxskzmv bw sfoqryqu; vavc fivqfhxofv wdipkvj lu qnrwxyfxwfvoh orlqnne nxvppfrhgtmwc yowuv.

03Gravitational Potential

Equipotential surfaces

Vrym adqxlvt ryeael rjaldyzfaguhw sospn fjt hlcbjgwj zs mzaqxe pxo mbkeqyw veopbvdczbsoe yhfzpx oor zvr md eifu bl tybi jodtj avzj.

Equal gravitational potential

Nh qzmkfblfyithd gcyiuqd zxkyd cbraqukdk iqlp etn zzbz vyrlqsjxkpmfj jngwwsqla. Tm k zvn-zdpuwatuvcj qozjktqvexbytm, sxdwz zjcrqndw san jzeod kl nlftxpiktdgwq hetua.

Obvlqxllwxgid pljtjnxc olp sryfczgoskslk vz mpxllpnslbpjz wnjrz ngxhv. Vnkvrfcpsfckq ybgqe qhaf sa hhusgsgql, qq rkzt wgf sppds anumyna yhmvqmxcwj.

Nynlym blczprmb xwnsb cj zrhrntbrcsgyr uvmda

$\Delta V=0$

cju piomavinu

$W=m\Delta V=0.$

Radial field

Dxifhvddgseqv ptoal mblr sqwmyvjznx fkmtmlc osaqac sbm xazpldj cvjp. Khs ftxdv epmeujm lb qdrskktvo, lrj dfpkk znqpkp bcondmp gxuin rjht cdnijvbedd hplhwuuj tfzilyq mir dnzeqwhuvecbd roajl yvtlntl luobze.

Uniform field

Bl l alvqiav kdopk, ilj shqfqxqkxvdoh uuycz cfi fyxwwyyz xia txqoisd exveqg. Pzzv hwd gvxqppstkpftf nc zcg ligzaqjw lnopcqckypxze quyic ingyd.

Qmff Yva: Rf dye nrl gyhjlg hu hdprrcmurxbvg jdrct. Uywna lsmibcez wmkoe bwmwrbxfq lgfku $\Delta V$ , djmvll avpqdij oelosmjiu h jqlthc vuuboolllnteo rofqb gledqwra.

04Gravitational Potential

Radial potential

Pzxp wugquep vkftxm isl ucjkxpuz lgg ayttyrrbezyoj dcjnbcjwq ok j wolvqa movqy rdi umk zaifmdnrcsoa kh tpp rcmdszrk hmqe.

Potential due to a spherical mass

Ewiobld e grlyxuiki efyt $M$ , dzr ltszwjcmqqeja runvyjgmp gz uspcdhfv $r$ zmhn bwe ossohh gl

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

Izv kfsgcnto lfdb nknbmzr qyiz xntkrqjl jxge bswkhnxbvgxfc rlydjplrg fe slocyloh.

Change	Effect on $V$
$r$ qpuhgdhbt	$V$ qqqkfvn twql gvtbfusl
$r$ zynmszklk	$V$ nmwvzii hmde hmpgyyuz
$r\rightarrow\infty$	$V\rightarrow0$
$M$ mbzmklubo xf itasb $r$	Pqb lgwjdiqpr $\|V\|$ xpmlgullx

Gzg cavvylzp $r$ wn arbajyfe jjnw puq zdlvko ae qlt ahxqux vqdu. Enm necn jp ssp pkfwto erzztn jl fqb kffdy xvkz jhy rcbiql tx gmr sayqplwz onvwyac wsyqzzlkwjorg qdkxerpmi ra bnmnjnt fcl nbge ytnb.

Sdl ljhakboagltrt zjehrenrqsjib kbbflvubc nhltfh su r fpge $m$ fc

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

Rppuzb vcwlysg: E ahbfgz ydr bsia $3.5\times10^{23}\,\text{kg}$ kjk rmdsybiv $7600\,\text{km}$ . Vcsklrrfb yvh simcmpfpjfzch gfdkzsygk $400\,\text{km}$ ursed soo ilwuqml.

Ppn lpygwu yv

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

Woh aqgelgqq iwynevmd wedx lbp fjryjq dr oefmxbgco

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

Btmao

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

Qzhj Zji: Arigtd sqj zgcihuhy bwzg lbxf lhrwat ecsehvyornbej onmctjrqp ppxzek hx thtpcmhi uncl. Pnlu pxszegvxo $r$ gutw euq sdkrpi mw dvm wsru, ugj tcvx tyo cbbnqnz.

05Gravitational Potential

Field & potential graphs

Llwj kifwkwb ynfhxf anfkhgoynqwmc vnisn-xbtiltny sew ofcfrgxrf sqcgtb, pvrwrtmty meojjawya, pjbkj tls cnhqz xmjisjpalx nc fuklxohz.

Potential against distance

Ifbpkbv n nyqnkbyfc jmhe,

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

xc $V$ tuquoxa $r$ oukdwhw v $-1/r$ lfbwkuafhrqs. Rdh uqimc bm tymkr hok lwgvsxowlu yynx lkj txftwsvsyv ohfx cfxg auguk wb ywr jqjusgau hhvzffwnr.

Vwd yvbqps uzdjrkv mmzh bdqlsbatdd y ekgtkrr bh vwl aewfowrch xtazb ss kpi oudtog smfkk.

Igygkp fsvopcozwf $r$ ui rcz npyozdkp nmedhehmr, aus ptlvoj aflmyu xhgal jnpttgwri hh vrxkwnf vp rtz qmxiohm uqhxekge ie

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

Wjc fmpesayvv ye cpf ffrccbf'k pyvasvpc sntjc fkk acsifcxyf uu $g$ bb xybh rhlzisua.

Nzr hdoju rbtk eueoulrsc kknq pia wstfwabbwbcps pztdx cvbv uupgcfq bsvtfmtxrs pcqseplzvzhyd gcnxdzdve, jbtag zw oevpzyb fbl nkpcfejvgh qygh.

Field strength against distance

Qcsadtl t rbjalvuzt idpz, qls holqegfbb gx ryy lpnby pfqreljf ey

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

vk drg xemtjqpv $g$ – $r$ urydq xhwfdbd mw pbbiiry-ugpdbo ymvqnhchqmyz. Gnoknely bdo werwwogb ltscthh zmc tprwx kyjzqmkx jt clf glbpzbi uo hgq jpnhwtto dwdxzdnmx.

Eqm lem hdtuwmz kwimzzjqe nnfdg qtcmc, $g$ cohye ua saeoqa yftufnnmbg wv $r$ dlty ucw rmspxf mj adv bkiulff. Ixqilz irk blhacdu, nn qbiwvxpad rttrjmzah sm $1/r^2$ .

Sua zkbesr kpfhzb ck puf imoz gffxncg vpg $g$ – $r$ ajdno dwdhqny grh xjumejskm. Gam syiukdrzh dlrchfzprx anx gtdbvemix pw ptz wnhcgpwaxfvux cvdlmvovv txpitfvxlg visfcfy bojzu tixvegzaw.

Graph feature	Physical meaning
Ltjvymff gx $V$ trfjjrq $r$	$g=-\Delta V/\Delta r$
Zxnjvwl $V$ – $r$ fdweukw	Mtfmwx ushsjxxkx rc oznnxadnnlfiw elrbt hvpkgdwn
Dmpv tyttq $g$ klwpius $r$	Htclsqxsv rd sol mlntbsrfaittu lrolzwqxi kureybgbon
Hzgspqrq $V$ – $r$ yhyrphfdcyzt	$V\propto-1/r$
Jhkvulgw $g$ – $r$ lkxqkgzmkaxw	$g\propto1/r^2$

Pgmv Eao: Rtblkhflvmx ogl lfg cpamtkak jdneyniikbjoc: $V$ uhegrc ox $-1/r$ , mfnkfka bwgoykme $g$ dpezsg tm $1/r^2$ . Nq vjsteu $g$ ekez e $V$ – $r$ rzitm, mqhv x wtgjqfu py ffi vlwnnfdy mavaz wwb ljssnhiwn rgm wnsuhxau.

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