Lecture 3: Resistive forces, and Energy

Transcription

1 Lecure 3: Resisive frces, and Energy Las ie we fund he velciy f a prjecile ving wih air resisance: g g vx ( ) = vx, e vy ( ) = + v + e One re inegrain gives us he psiin as a funcin f ie: dx dy g g = vx, e = + v + e d d g g x ( ) = x + vx, 1 e y ( ) = y + v + 1 e g g x, x ( ) = v e + C y ( ) = v + e + C

2 Le s again find he range f he prjecile Raher han wriing y in ers f x, we can slve fr he ie f fligh, and plug ha value in he equain fr x g g y ( ) = y = y + v + 1 e 1 g = + g v 1 e This is a ranscendenal equain, eaning here is n exac sluin Hwever, as physiciss, we re n allwed give up! We need find he bes apprxiae sluin ha we can Of curse, i s easy d n a cpuer, bu if we re suc n a desered island (r ding P31 hewr).

3 Perurbaive Sluin (fr Sall /) If we assue ha / is sall, we can use a Taylr expansin find an apprxiae answer fr : 3 1 g 1 1 = v g 6 1 g 1 1 v + + g 6 3 g 1 1 = g + 6 v g 1 v 1 = 1 + g = + 6 g 6 v v + +

4 1 g 1 v 1 g 6 ( ) v 6 v + g + v 1 v / g 1 = 1 + = + = + = + v + g 3 v / g The firs er has he expansin paraeer in he deninar, s i als needs be expanded: v v 1 = g + g v v = + g 3 g Ne ha if is zer, we ge exacly he answer we had befre cnsidering air resisance (if we didn, we d need chec fr errrs!) The equain fr is quadraic, which we can slve exacly

5 Hwever, we can ge a gd apprxiain by aing he assupin ha he he sluin has he fr: where 1 is sall = + Plugging his in ur equain gives: ( 1 ) + v + 1 = + 3 g + 3 v g 1 ( v / g ) v v = + g 3 g v v v v = 1 g + = 3g g g v 1 3 g

6 The las sep, where assued ha he sluin is nly slighly differen fr an answer we already nw, is he disincive feaure f he apprxiain echnique called he perurbaive ehd This echnique is very ipran hrughu physics Yu are sure see i again in Quanu Mechanics In fac, in wha we currenly believe is he crrec descripin f naure, quanu field her we are unable exacly slve he equains fr anyhing Perurbaive echniques are ur bes eans f arriving a precise nuerical predicins ha can be esed in experiens (hugh se researchers are aing grea srides in using cpuers find apprxiae sluins)

7 Wr and Energy We define he fllwing quaniy as wr: W r = r 1 dr This is an exaple f a pah inegral I sees lie an dd defniin, bu i has ne very ineresing prpery: r dv dv dr dr = dr = d d d d r r r ( v v) dv 1 d = v d = d d d v = d v = v v v 1

8 1 If we nw define he quaniy as he ineic energy (T), we have: v W = T This is he wr-energy here This prvides a way deerine he change in velciy f an bjec wihu inegraing he equains f in In he definiin f wr, is he al frce acing n he bjec Nw cnsider he case where uliple frces are acing: fr exaple, iagine lifing a bx fr he flr a shelf There are w frces a wr graviy and he frce supplied by he lifer If he bx is lifed a cnsan velci here is n change in ineic energy in he prcess

9 Des his agree wih he wr-energy here? Yes! The ne frce acing n he bx is 0, s bu i als eans he wr-energy here isn very useful in his case Bu can we say anyhing ineresing abu he wr dne by he graviainal frce alne? rce as a funcin f psiin is a vecr field = gj fr gravi i s W = d r = 0 = T In his case, we can rewrie he frce in ers f a scalar field: = ( gy) j = ( gy) y

10 If we call his scalar field U, we can express he wr dne by graviy as: W = dr = U dr = U We call U he penial energy assciaed wih he graviainal frce Ne ha i has he sae unis as ineic energy Ne als ha he value f he pah inegral in his case depends nly n he endpins f he pah, n n he pah iself i.e., all f he fllwing pahs have he sae d r 1

11 When can we find a penial energy? rces fr which he wr dne beween w pins is independen f he pah aen are called cnservaive = U The requireen ha he frce be f he fr is an equivalen definiin Penial energy can nly be defined fr cnservaive frces Bu hw can we ell wheher a given frce is cnservaive? Ne ha: ( U ) ε ( U ) = ij i = ij ij x j ij x j x e ε U Defined as: ε13 = ε31 = ε31 = 1 ε = ε = ε = 1 ε e ij i = 0 if any index is repeaed

12 Using he definiin f ε ij we see ha U U ( U ) = e i = ij x j x x x j 0 Any frce which has a curl equal zer is cnservaive T ge an inuiive picure, a frce wih a nn-zer curl has lines f frce ha fr clsed lps: As a paricle ravels arund his lp, he wr dne by cninues increase Penial energy is n eaningful

13 When Is Energy Useful? Clearl if nly cnservaive frces ac n a paricle, energy can be used slve prbles Hwever, even if here are nn-cnservaive frces, energy is sill useful if he nn-cnservaive frces dn d any wr Exaple: An bjec sliding dwn a fricinless rap f arbirary shape Bh graviy and he nral frce N ac n he bjec N is n cnservaive Bu since, he N dr = 0 nral frce des n wr Can find speed as a funcin f heigh using energy