Problm : Journy to th Cntr of th Earth Imagin that on drilld a hol with smooth sids straight through th ntr of th arth If th air is rmod from this tub (and it dosn t fill up with watr, liquid rok, or iron from th or) an objt droppd into on nd will ha nough nrgy to just xit th othr nd aftr an intral of tim Your goal is to find that intral of tim a) Th graitational for on an objt of mass m, loatd insid th arth a distan r < R, from th ntr (R is th radius of th arth), is du only to th mass of th arth that lis within a solid sphr of radius r What is th graitational for as a funtion of th distan r from th ntr? Exprss your answr in trms of g and R Not: you do not nd th mass of th arth to answr this qustion You only nd to assum that th arth is of uniform dnsity b) Using th onpt of for xplain how th objt-arth systm for objts insid th arth is analogous to an objt-spring systm ) What is th potntial nrgy insid th arth as a funtion of r for th objt-arth systm? Can you think of a natural point to hoos a zro point for th potntial nrgy? Find out how long this journy will tak d) How dos this tim ompar to a satllit in low arth orbit (i orbits just abo th surfa of th arth)? ) In ordr to shortn th tral tim, th sndr trals down th hol a distan until sh is only a fration of th radius of th arth, β R, (R is th radius of th arth) from th ntr Th ripint trals down a similar distan from th far nd in ordr to ath th objt By how muh if any is th tral tim thrby rdud (as a funtion of β)?
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Dpartmnt of Physis Physis 801 TEAL Fall Trm 004 In-Class Problms -4: Harmoni Osillation and Mhanial Enrgy: Solution Problm : Journy to th Cntr of th Earth Imagin that on drilld a hol with smooth sids straight through th ntr of th arth If th air is rmod from this tub (and it dosn t fill up with watr, liquid rok, or iron from th or) an objt droppd into on nd will ha nough nrgy to just xit th othr nd aftr an intral of tim Your goal is to find that intral of tim a) Th graitational for on an objt of mass m, loatd insid th arth a distan r < R, from th ntr (R is th radius of th arth), is du only to th mass of th arth that lis within a solid sphr of radius r What is th graitational for as a funtion of th distan r from th ntr? Exprss your answr in trms of g and R Not: you do not nd th mass of th arth to answr this qustion You only nd to assum that th arth is of uniform dnsity Answr: Choos a radial oordinat with unit tor rˆ pointing outwards Th graitational for on an objt of mass m at th surfa of th arth is gin by two xprssions Gmm F = gra rˆ = mgrˆ R Thrfor w an sol for th graitational onstant alration 1
Gm g = R Whn th objt is a distan r from th ntr of th arth, th mass of th arth that lis outsid th sphr of radius r dos not ontribut to th graitational for Th only ontribution to th graitational for is du to th mass nlosd in th sphr of radius r, Sin th mass dnsity is gin by 4 m nlosd = ρ π r ρ = m (4/)π R, th mass nlosd is m m nlosd = (4/)π r = m r (4/)π R R Thrfor th graitational for on th objt of mass m whn it is a distan r from th ntr of th arth is gin by = Gmm nlosd rˆ = Gmm r Gmm Fgra rˆ = r r r R R W an us our xprssion for th g Gm = / R to find that th graitational for on a mass m at a distan r from th ntr of th arth is gin by F = mg rrˆ gra R b) Using th onpt of for xplain how th objt-arth systm for objts insid th arth is analogous to an objt-spring systm Answr: Th minus sign indiats that th for is always dirtd towards th ntr of th arth (rstoring for) and proportional tot h distan from th ntr of th arth This is analogous to th rstoring for of a spring, F spring = kxxˆ,
whr th spring onstant for graitation is gin by k = mg gra R This mans that th objt will undrgo simpl harmoni motion just lik a spring ) What is th potntial nrgy insid th arth as a funtion of r for th objt-arth systm? Can you think of a natural point to hoos a zro point for th potntial nrgy? Find out how long this journy will tak Answr: W an dfin a potntial nrgy for graitation insid th arth that is analogous to th spring potntial nrgy funtion with zro point for potntial nrgy hosn at th ntr of th arth and 1 Ur () = k gra r = 1 mg r R If w rlas th objt from rst at th surfa of th arth, th initial mhanial nrgy is all potntial nrgy and is gin by \ E = ( ) = 1 mg R = 1 i U R mgr R Whn th objt rahs th ntr of th arth, th mhanial nrgy is all kinti nrgy, E f = K f = 1 m Sin thr ar no xtrnal work ating on th systm, th mhanial nrgy is onstant and E i = E f 1 1 mgr = m Th loity just bfor th objt rahs th ntr of th arth is thn = gr
Not w hoos just bfor so that th loity is radially inward in polar oordinats: thr is no wll dfind dirtion whn th objt is loatd at th origin From Nwton s Sond Law, F = ma, in th radial dirtion boms This simplifis to Th position of th objt satisfis d r rˆ : mg r = m R dt dr g = r dt π rt () = R os( t) T whr T is th priod of osillation Th loity of th objt is R π π t () = R sin( t) T T π π Whn t = T /4, os( t) = os( ) = 0 hn th objt is at th ntr of th arth Also T π π sin sin( t) = sin( ) = 1, th loity at th ntr of th arth is gin by T π = ( T / 4) = R T From th ondition that mhanial nrgy is onstant w found that = gr Thrfor π R = gr T W an sol for th priod and find that T = π ( R / g 1/ ) d) How dos this tim ompar to a satllit s priod in low arth orbit (i orbits just abo th surfa of th arth)? 4
Answr: If an objt undrgos uniform irular motion around th arth th Nwton s Sond Law in th radial dirtion boms π rˆ : mg = mr T W an sol for th priod of th irular orbit and find that th priod is idntial to th priod of osillation, T = π (R / g) 1/ ) In ordr to shortn th tral tim, th sndr trals down th hol a distan until sh is only a fration of th radius of th arth, β R, (R is th radius of th arth) from th ntr Th ripint trals down a similar distan from th far nd in ordr to ath th objt By how muh if any is th tral tim thrby rdud (as a funtion of β)? Answr: If th objt starts losr to th arth th priod is still th sam baus w ha only hangd th initial onditions w ha not hangs th spring onstant of graitation 5