12 Gravitational Force Near the Surface of the Earth, First Brush with Newton s 2 nd Law

Transcription

1 Chpter Grvittionl Force er the Surfce of the Erth, Firt Bruh with ewton nd Lw Grvittionl Force er the Surfce of the Erth, Firt Bruh with ewton nd Lw Soe folk think tht every object ner the urfce of the erth h n ccelertion of 9.8 / downwrd reltive to the urfce of the erth. Tht jut in t o. In fct, I look round the roo in which I write thi entence, ll the object I ee hve zero ccelertion reltive to the urfce of the erth. Only when it i in freefll, tht i, only when nothin i touchin or puhin or pullin on the object except for the rvittionl field of the erth, will n object experience n ccelertion of 9.8 / downwrd reltive to the urfce of the erth. Grvittionl Force ner the Surfce of the Erth We ll live in the inviible rvittionl field of the erth. M i lwy ccopnied by urroundin rvittionl field. Any object tht h, includin the erth, i urrounded by rvittionl field. The reter the of the object, the troner the field i. The erth h hue ; hence, it crete tron rvittionl field in the reion of pce round it. The rvittionl field i force-per- t ech nd every point in the reion round the object, lwy redy nd ble to exert force on ny prticle tht find itelf in the rvittionl field. The erth rvittionl field exit everywhere round the erth, not only everywhere in the ir, but out beyond the tophere in outer pce, nd inide the erth well. The effect of the rvittionl field i to exert force on ny prticle, ny victi, tht find itelf in the field. The force on the victi depend on both property of the victi itelf, nely it, nd on property of the point in pce t which the prticle find itelf, the force-per- of the rvittionl field t tht point. The force exerted on the victi by the rvittionl field i jut the of the victi tie the force-per- vlue of the rvittionl field t the loction of the victi. Hold rock in the pl of your hnd. You cn feel tht oethin i pullin the rock downwrd. It cue the rock to ke teporry indenttion in the pl of your hnd nd you cn tell tht you hve to pre upwrd on the botto of the rock to hold it up int tht downwrd pull. The oethin i the field tht we hve been tlkin bout. It i clled the rvittionl field of the erth. It h both nitude nd direction o we ue vector vrible, the ybol to repreent it. In enerl, the nitude nd the direction of rvittionl field both vry fro point to point in the reion of pce where the rvittionl field exit. The rvittionl field of the erth, ner the urfce of the erth i however, to very ood pproxition, uch ipler thn tht. To very ood pproxition, the rvittionl field of the erth h the e vlue t ll point ner the urfce of the erth, nd it lwy point towrd the center of the erth, direction tht we norlly think of downwrd. To very ood pproxition, = 9.80 downwrd (-) k t ll point ner the urfce of the erth. The fct tht the rvittionl field i force-per- t every point in pce en tht it ut hve unit of force-per-. Indeed, the (newton) 69

2 Chpter Grvittionl Force er the Surfce of the Erth, Firt Bruh with ewton nd Lw pperin in the vlue i the SI unit of force (how tron the puh or pull on the object k i) nd the k (kilor) i the SI unit of, o the /k i indeed unit of force-per-. The rvittionl force exerted on n object by the erth rvittionl field (or tht of nother plnet when the object i ner the urfce of tht other plnet) i oetie clled the weiht of the object. To tre tht the rvittionl force i force tht i bein exerted on the object, rther thn property of the object itelf, we will refer to it the rvittionl force. The rvittionl force F exerted on n object of by the rvittionl field of the erth i iven by F = (-) The product of clr nd vector i new vector in the e direction the oriinl vector. Hence the erth rvittionl force i in the e direction the rvittionl field, nely downwrd, towrd the center of the erth. The nitude of the product of clr nd vector i the product of the bolute vlue of the clr nd the nitude of the vector. [Recll tht the nitude of vector i how bi it i. A vector h both nitude (how bi) nd direction (which wy). So for intnce, the nitude of the force vector F = 5 downwrd, i F = 5 newton.] Hence, F = (-3) relte the nitude of the rvittionl force to the nitude of the rvittionl field. The botto line i tht every object ner the urfce of the erth experience downwrd-directed rvittionl force whoe nitude i iven by F = where i the of the object nd i k When the Grvittionl Force i the Only Force on n Object If there i non-zero net force on n object, tht object i experiencin ccelertion in the e direction tht net force. How uch ccelertion depend on how bi the net force i nd on the of the object whoe ccelertion we re tlkin bout, the object upon which the net force ct. In fct, the ccelertion i directly proportionl to the force. The contnt of proportionlity i the reciprocl of the of the object. F (-4) Eqution -4 i known ewton nd Lw. ewton nd dp Lw cn lo be written F = where p i the dt oentu of the object. Thi ltter eqution i vlid t ll poible peed, even t peed cloe to the peed of liht. To derive eqution -4 fro it, we ue p = v which i only vlid for peed ll copred to the peed of liht. Hence, eqution -4 i only vlid for peed ll copred to the peed of liht ( /). Alo, 70

3 Chpter Grvittionl Force er the Surfce of the Erth, Firt Bruh with ewton nd Lw The expreion F en the u of the force ctin on the object. It i vector u. It i the net force ctin on the object. The i the inerti of the object, the object inherent reitnce to chne in it velocity. (Inherent en of itelf. ) ote tht the fctor in eqution -4: F en tht the bier the of the object, the ller it ccelertion will be, for iven net force. Eqution -4 i concie tteent of ultitude of experientl reult. It i referred to ewton nd Lw. Here, we wnt to pply it to find the ccelertion of n object in freefll ner the urfce of the erth. Whenever you pply ewton nd Lw, you re required to drw free body dir of the object whoe ccelertion i under invetition. In free body dir, you depict the object (in our ce it i n rbitrry object, let think of it rock) free fro ll it urroundin, nd then drw n rrow on it for ech force ctin on the object. Drw the rrow with the til touchin the object, nd the rrow pointin in the direction of the force. Lbel the rrow with the ybol ued to repreent the nitude of the force. Finlly, drw n rrow ner, but not touchin, the object. Drw the rrow o tht it point in the direction of the ccelertion of the object nd lbel it with ybol choen to repreent the nitude of the ccelertion. Here we ue the ybol for the ccelertion to reind u tht it i the ccelertion due to the erth rvittionl field. Free Body Dir for n Object in Freefll ner the Surfce of the Erth F The next tep in pplyin ewton nd Lw i to write it down. = F (-5) the concept of force prove iprcticl on the toic cle nd ller (ditnce le thn bout 0-9 ). Such ll cle re the rel of quntu echnic where enery nd oentu till ply jor role. 7

4 Chpter Grvittionl Force er the Surfce of the Erth, Firt Bruh with ewton nd Lw ote tht eqution -4: F i vector eqution. A uch it cn be conidered to be three eqution in one one eqution for ech of totl of three poible utully-orthoonl (enin perpendiculr to ech other) coordinte direction in pce. In the ce t hnd, ll the vector, (hey, there re only two, the rvittionl force vector nd the ccelertion vector) re prllel to one nd the e line, nely the verticl, o we only need one of the eqution. In eqution -5, = F we ue rrow ubcript the rrow hft linent pecifie the line lon which we re uin the force nd the rrowhed pecifie the direction lon tht line tht we chooe to cll the poitive direction. In the ce t hnd, referrin to eqution -5, we note tht the hft of the rrow re verticl, enin tht we re uin force lon the verticl nd tht we re delin with n ccelertion lon the verticl. Alo in eqution -5, we note tht the rrowhed re pointin downwrd enin tht I hve choen to cll downwrd the poitive direction, which, by defult, en tht upwrd i the netive direction. (I choe to cll downwrd poitive becue both of the vector in the free body dir re downwrd.) ext we replce with wht it i in the free body dir, F nely, nd we replce F with the u of the verticl force in the free body dir, countin downwrd force poitive contribution to the u, nd upwrd force netive contribution to the u. Thi i n ey ubtitution in the ce t hnd becue there i only one force on the free body dir, nely the rvittionl force, the downwrd force of nitude F. The reult of our ubtitution i: = (-6) F The F in eqution -6 i the nitude of the rvittionl force, tht force which you lredy red bout t the trt of thi chpter. It i iven in ter of the of the object nd the nitude of the erth rvittionl field by eqution -3, F =. Replcin the F in eqution -6, = F, with the to which it i equivlent we hve 7

5 Chpter Grvittionl Force er the Surfce of the Erth, Firt Bruh with ewton nd Lw = (-7) ow the tht pper in the frction i the inerti of the object. It i the ount of inherent reitnce tht the object h to chne in it velocity nd i eure of the totl ount of teril kin up the object. The pperin in the prt of the expreion (eqution -7) i the rvittionl of the object, the quntity tht, in concert with the rvittionl field t the loction of the object deterine the force on the object. It i lo eure of the totl ount of teril kin up the object. A it turn out, the inertil nd the rvittionl of the e object re identicl (which i why we ue one nd the e ybol for ech) nd, in eqution -7, they cncel. Thu, = (-8) ow i the nitude of the erth rvittionl field vector t the loction of the object. = 9.80 /k nd, bein n ccelertion h to hve unit of ccelertion, nely, /. k Fortuntely newton i Thu o the unit of, nely /k, do indeed work out to be. = 9.80 (-9) ow thi i wild! The ccelertion of n object in freefll doe not depend on it. You w the e cncel. The e thin tht ke n object hevy ke it luih. One-Dienionl Free-Fll,.k.., One-Dienionl Projectile Motion If you throw n object triht up, or iply relee it fro ret, or throw it triht down; uin tht the force of ir reitnce i neliibly ll copred to the rvittionl force: the object will be in freefll fro the intnt it loe contct with your hnd until the lt intnt before it hit the round (or whtever it doe eventully hit), nd the object will trvel lon triht line pth with contnt ccelertion of downwrd. Conider the ce in which the object i thrown triht up. The whole tie it i in freefll, the object experience n ccelertion of downwrd. While the object i on the wy up, the downwrd ccelertion en tht the object i lowin down. At the top of it otion, when the velocity chne fro bein n upwrd velocity to bein downwrd velocity, nd hence, for n intnt i zero, the downwrd ccelertion en tht the velocity i chnin fro zero to non-zero downwrd velocity. And on the wy down, the downwrd ccelertion en tht the velocity i increin in the downwrd direction. 73