07 Gravitation Notes

P a g e 1 4 of 1 9 T he resultant field strength at a point due to m ore than one m ass can be found by the T he gravitational field strength at a point is the acceleration of free fall at that point. created it. G r av itat ion a l fiel d stren gt h is a vec tor qu a n tity an d it po i nts tow a r ds the ma s s wh i ch N ote Sh. II : e g. A lH fut ' "lwomMp(ht. ' " lsglm by ¡Æ ¡¤ ªÆ产 国ì×国 ¡£ * N x a©¥\h ߣ诈 uniform with a m agnitude of 9 . 8 1 N kg - 1 . H ence, n e a r the surface of the E aM , w e c a n c o n sider the gravitationa©¥ tield to be Ù³to be parallel to one another as shown in the figure on the right. lts centre. If we zoom In to a region near the surface of tFre E arth, the field linea e e T he figure on the left show s the field lines around E arth, w hich are directed towarda centre. F or a po©¥nt m ass or a uniform apherica©¥ m ass. The field ©¥ines are directed toward©¥ ll©¥ gravitational fie©¥d atrength wil©¥ have do©¥er or denaer field lines, a n d vice versa. T he dene©¥tv of Êö ©¥e neid linea indica©¥es ©¥t& l . A region with a ©¥©¥ronger T he direction of the neid línea ©¥ndicales the ģ clion Q t M Q i o n a ©¥ ©¥iau , gravitational force. F©¥e©¥d L ©¥nes fie©¥d ©¥lnea ©¥n which the tangent at a point on a tield ©¥ine ©¥ct©¥ ©¥n the direction ol the G rav©¥tat©¥onal A reg©¥on of gravitational neld can be vi©¥uali©¥ed as con©¥i©¥©¥lng of an ©¥rr ¡¤ y of Im ©¥glnary , - ~ " J» : Į r enceB a gravi©¥allona©¥ force ©¥là w av©¥tat©¥ona©¥ neid ©¥©¥' region of ©¥p©¥ce ©¥n w hich . M ass p©¥aced ©¥n that reglon©¥F©¥eld G rayltational 7 . 2 G ravitational F l ¡¤ ©¥d S trength g Y E A R S - ©¬ pH Y S ©¥C S D E P A R T M E N T R A FFL E S ©¥N ST ©¥TU T©¥O N

P a g e ©¥6 of 1 9 Q uestion W hat happens if the E arth spins faster about its axis of rotation? object m oving in a uniform circular m otion. A t the E quator, part of the gravitationa©¥ force provides for the centripetal force to keep the value is know n as the apparent w eight. T h e spr ing ba l an c e indi cates a value les s tha n the true we i gh t of the ob ject. T hisa T herefore, T = F . - F c . S in ce th e o bject undergoes circu©¥ar m otion, F R - F c - F . - T A pplying N ewton ' s s e c o n d law , Fn - F . - T R esultant force F n at the E q u ato r T h e sp r ing ba l an c e indi cates F. . wh ich is the true we ight of the object. T herefore, T = F , S ince there is no circular m otion, Fn - O: $ Ply" g N ewton ' " " ' " law , F n - F * - T R esultant force F at the polar region m otion at the E quator. and po©¥ar regions. H owever, T wil©¥ be di¨¤erent because the m ass undergoes circular lf the E arth is taken to be a unworm sphere, then F ©¥i the sam e at both the equator©¥a©¥ ror o \ :Ť L ) R otation w eight) and force T due to the apring . due to E arth ' = ©¥n b o th c a s e s ©¥h e re a re ©¥w o fo rce ©¥ a clln g o n m a s s - gravi©¥ationa©¥ force F , (true F ield S trength G ravltat©¥onal the E arth - the E quator and the N orth P ole. V ariation ©¥n T he figure showa an object or m ass m hanging trem a spring balance at ©¥w o places on Y EA R 5 - 6 P H Y S IC S D E P A R T M E N T R A F F L E S ©¥N S T ©¥T U T ©¥O N

P a g e ©¥7 of 1 9 \ ų tu lį' ; . \ . TmwitlmJ m inerals and . ' ï . ' gravitational field caused by m ountains and trenches, a s w e ll as the presence of T he E arth ' s ìŝ ©¤oi unita . T here are local variations in the E arth ' s 2 . Fie©¥d S trength G ravitational diam eter. A ¨¤ecting 6 a l sym©¥ lts polar diam eter ls about 4 0 km lesa than its equatorial other F actors 1 . T he gravitational fie©¥d strength over the E arth ' s s u rface varies as the reault of its©¥i Ï¡ÙÍ » t o\ S olut©¥Q n brn? r read In kg? (©¥©¥) If a 6 . 0 kg m asa ©¥©¥ p©¥aced on a weighing = ca©¥e at the E quator, w hat wou©¥d the eca©¥e object placed at the E quator. (©¥} G iven that the E arth ' a ra dius la 6 3 7 o km , find the centripeta©¥ acce©¥eration of an E xam p©¥e 7 . 4 Y E A R S - 6 P H Y S IC S D E P A R T M E N T R A F F L E S ©¥N S T ©¥T U T IO N

©¤r¨© - - " - pa g e 1 8 of 1 9 O bjects w hich are bounded have a negative total energy! T he centripetal force is pr «í vided by the gravitational force due to the planet. S ince satellite is m oving in a uniform circular m otion, it experiences a centripeta©¥ force. ş olu!ipņ W hat is the tota©¥ energy of the satellite? m otion of radius r around a p©¥anet of m ass M . A satellite of m ass m is m oving in a uniform circularE xam ple 7 . 5 ¡ß

P a g e 1 9 of 1 9 adding of the potentials at that point due to each of the m ass. T he gravitational potential at a point due to two or m ore m asses can be found by T he negative sign is not an indication of direction and it cannot be om itted. G ravitational potential is a scalar quantity . N ote negative value H ence, based on its definition, gravitational potential (or potential energy) has a T h i s res u ©¥ts in ne g a t ive wo r k do n e by the ex ter na ©¥ for ce. m ass as it is being low ered tow ards the E arth, ) of dlsp©¥acė m ent of the m ail (w e can im agine the external force supporting the in the gravitational lgrect©¥on of the extem a©¥ e©¥ is Đ pğ to the H ©¥iettïm S ince gravitationa©¥ force is attractive in nature, to briN a m ass from Infinity to a point G r av©¥tationa©¥ poten tla©¥ (or potentia©¥ energy) at ©¥nfinlly ©¥a zero. A Thy ©¥ ¡¤ grav©¥tat©¥ona©¥ potent©¥a©¥ (or potent©¥©¥©¥ energy) nogat©¥ve In v©¥©¥ue? 4 - ￠ f h * m r f i a ©©hto卤ì£òæì£ìÑ 囁 rų rf©¥u where the m ass m of the em a©¥©¥ test m ass cancels ou©© t. ' " J m W r ¡¤ \m Į ©¥ d . , " " t" " k" " oi" " aa ĺ '" . . - ￠ - ii P otent©¥a©¥ ' c©¥©¥n « m » W =T a¡¿tem al force in bringing a sm al©¥ leal m ass from infinity to rìi¨¤rlm G ravitationa©¥ liiiai©¥ at a poi nt ©¥n a g ra v it a©¥ ion a©¥ file Id la d efined a s tr©¥e I Y E A R S - 6 P H Y S IC S D E P A R T M E N T R A F F L E S IN S T FRU T IO N

P a g e lo of 1 9 ' ttl q , m W hat is the w ork done w hen a 2 kg m ass is taken from X to Y ? 2L L T he gravitational potential at X is - 8 kJ kg - 1 . T he figure show s tw o points X and Y at distances L and 2 L from the centre of the E a E xam p©¥e 7 . 7 tS A JC ©¥H 2©¥P re©¥©¥m s 2 0 1 0 /p1 ©¥1 3 - m o dified] 1, 18¡¿1 0 + 1 kg- 1 ==== 6 . 6 7777 ¡¿ll11 0000 - 1111 1111 ¡¿ 5555 . 9 8 ¡¿ 1 0 24 (īōir - ã3ã3ã38 12 0 012 0 0)))) - G m (i ©©) E arth to a point 1 2 0 0 m above the surface . the change in the gravitational potentia©¥ as an object ls m oved » om the surface of the G iven that the m ass of the E arth is 5 . 9 8 ¡¿ 1 o 2 4 k g a n d its ra d iu s is 6 . 37 ¡¿ m 6 m , c a ©¥cuïateE xam p©¥e 7 . 6 Y E A R 5 4 P H Y S ©¥C S D E P A R T M E N T R A F F L E S ©¥N S T ©¥T U T IO N