Discussion Introduction P1, Week 1 The Scientist s Sith Sense As scientist or engineer, uch of your job will be perforing clcultions, nd using clcultions perfored by others. You ll be doing plenty of tht in your clsses. But here s the difference: in clssroo eercises, the correct nswer is known in the rel world, it isn t! It s your responsibility to ke sure your work is correct (the consequences y be dire indeed if it isn t). Tody we ll work on developing tht sith sense tht eperienced scientists possess: they re constntly (lost subconsciously) checking their work ginst their physicl intuition. Here s the secret: Knowing wht the nswer will look like before you strt. And here re three guidelines tht becoe second nture to ny scientist: Drw sketch You should never strt work on proble without sketching the sitution first. A good sketch engges your physicl intuition nd often llows you to figure out 90% of the nswer before you write down single forul. Check your units If you re clculting force nd you get result tht coes out in eters, you know you de istke. If ny prt of your clcultion contins n epression like +v where is ss nd v is velocity, you know you de istke. Tht s ied units -- you cn t dd pples nd ornges, or ke the equl to ech other. Check liiting cses All probles hve liiting cses where you know wht the solution should be. E.g., run the sses or distnces in your proble to zero or infinity nd ke sure your solution gives the correct result. Let s try it out! Here re two siple eples to illustrte: () You re structurl engineer hired by n rchitecturl fir to ssist on the design of useu. In the plns is decortive corridor of length L whose roof will be supported by N coluns. ore coluns cost ore oney, but you hve to ke sure there re enough of the to support the roof sfely. You sk your ssistnt to clculte the ss tht ech colun will hve to support, depending on the design preters N nd L. Your ssistnt coes up with this forul: = C N / L where C = 3478.7 kg. Do you believe it? L (b) You re n strophysicist studying the otion of eteor in the grvittionl field of distnt plnet. The eteor hs ss, the plnet hs ss, nd the distnce between their centers is denoted r. The plnet hs cople ring structure your grdute student runs coputer siultion nd coes up with this forul for the force eperienced by the eteor: F = G (36.7 + 16.1 ) / r 3 where G is the good old grvittionl constnt. Do you believe it? r
Discussion Question 1A P1, Week 1 P11 Review: -D otion with Unifor Force The thetics nd physics of the proble below re siilr to probles you will encounter in P1, where the force is due to the ction of n electric field on chrged prticle. A point prticle of ss trvels freely in the -direction with unifor velocity v 0. At = 0, it enters region between two pltes oriented perpendiculr to the y-is; the plte spcing is w, nd then plte length in the -direction is L. The prticle enters on the id-plne y = 0. While between the pltes, it eperiences constnt, sptilly unifor force F in the +y-direction. After eiting the pltes the prticle gin oves freely. v 0 +w/ y L F w/ () Our first tsk will be to obtin n epression for the y-coordinte of the point t which the prticle eits the pltes. We will ssue tht the plte spcing is wide enough tht the prticle never strikes either plte. But before we strt, consider these possible solutions: FL (1) y (3) y Fw F () y (v0 L) v 0 v 0 Could ny of the be correct? Why or why not? Reeber, units nd liiting behvior! In fct, fro only those two considertions, you cn write down the correct nswer to within fctor of without using ny foruls t ll. Wnt to give it try? (This procedure is clled diensionl nlysis nd physicists use it ll the tie, especilly when developing new theories.) pge 1
(b) Now go hed nd clculte the correct epression for y. (c) Net, find n epression for the iu vlue of the force F for which the prticle psses through the force region without striking either plte. (d) For the conditions of prt (c), find n epression for tn where is the ngle of deflection t which the prticle eits the force region. (Did you drw sketch? Did you check your units?) pge
Discussion Question 1B P1, Week 1 P11 Review: Grvittionl Forces nd Superposition F 1 G 1 r 1 ˆ r 1, 1 r 1 ˆ r 1 This proble develops skills you will need in P1 in finding the electric fields creted by sets of point chrges. y X Four prticles of equl ss re fied t the corners of squre with sides of length. A fifth prticle hs ss nd oves under the grvittionl forces of the other four. () Find the - nd y-coponents of the net grvittionl force on due to the other four sses when is locted t the center of the squre (left-hnd figure). Hint: Drw sketch! Use superposition nd drw vector digr consisting of four vectors, ech representing the force eerted by one of the corner prticles on. For ese of reference, lbel the four (equl) corner sses 1,, 3, nd 4. Lbel the corresponding force vectors F 1, F, etc. With the vector digr in hnd, it is vstly esier to clculte the requested coponents of the totl force. pge 1
(b) Find the - nd y-coponents of the net grvittionl force on when it is locted t the center point of the right-hnd side of the squre (iddle figure). Use the se solution procedure tht ws recoended bove for prt (). (c) Check your nswer to prt (b) by testing t lest 3 eples of liiting behvior. Do you get the results you epect? Now, n iportnt point: suppose you were given nuericl vlues in this proble: = 3 kg, = 1 kg, nd = 5 c. If you hd plugged those nubers into your equtions right fro the strt, you d get the finl result F = 1.15 10-7 N. Would you be ble to check the liiting behvior of this nswer? No! We ve lerned n iportnt lesson: Never plug in nubers until the end of your clcultion. (d) When ss is locted on the -is distnce X lrge copred to (right-hnd figure), one cn use siple physicl rguent to see tht the net force on due to the other four prticles is pproitely F CGX nd F y 0 where C is nuericl constnt. (This is long-distnce pproition: it gets better nd better s X increses.) Wht is the pplicble physicl rguent? Use it to find the vlue of the diensionless constnt C. pge
Discussion Question 1D P1, Week 1 P11 Review: Unifor Circulr otion F 1 G 1 r 1 ˆ r 1, 1 r 1 ˆ r 1 In P1 you will encounter probles where chrged prticles ove in unifor circulr otion. The forces involved y be electric or gnetic in nture. The nswer to prt(e) contins the secret of the cyclotron. Kepler s Third Lw (K-III) for plnetry otion bout the sun for circulr orbits is T CR 3 where T is the period, R is the rdius of the plnet s orbit nd C is constnt. v R () Derive K-III for circulr orbit nd in the process find n lgebric epression for C in ters the ss of the sun S, the universl grvittionl constnt G, nd nuericl fctors. (b) Using your nswer fro prt (b), re-epress K-III s reltionship between the ngulr frequency of the otion nd the rdius R of the for: = f(r, G, S ). pge 1
(c) Consider unifor circulr otion of body of ss bout centrl force which depends on velocity: F Dv R b where D is constnt nd nd b re known eponents. Derive K-III for this force, gin epressing your nswer s reltionship between nd R, (The constnt D will necessrily pper in your finl epression, but v ust not.) (d) For the cse = 0 nd b =, verify tht your nswer to prt(c) collpses to tht for prt (b) (this is n ecellent liiting behvior check). (e) Evlute your nswer to prt (c) for the cse = 1 nd b = 0. How does the ngulr frequency depend on rdius for force of this nture? pge
Discussion Question 1C P1, Week 1 Review: Potentil Energy Four prticles of equl ss re fied t the corners of squre with sides of length. A fifth prticle hs ss nd oves under the grvittionl forces of the other four. Consider the grvittionl potentil energy U of the ss s function of its position (,y). This potentil energy p U(,y) is n etreely useful wy of representing the effect of the grvittionl force on the ss : If is held t rest t soe point ( 1,y 1 ) nd you let it go, it will lwys ove towrd point of lower potentil energy. You cn therefore think of the function U(,y) s topogrphicl p: the prticle will lwys roll downhill. If the ss oves fro point ( 1,y 1 ) to point (,y ), the potentil energy difference U( 1,y 1 ) U(,y ) tells you ectly how uch work ws done by grvity, nd ectly how uch kinetic energy the prticle gined s result. Let s visulize ll this. The figures below show the potentil energy p U(,y) for the squre configurtion of four sses illustrted bove. The height of the surfces indictes the gnitude of U t ech point. Which one is correct? 1
Now let s perfor soe clcultions using potentil energy. Here s the ster forul for the grvittionl potentil energy between two sses 1 nd seprted by distnce r 1 : U 1 G 1 r 1 () Show tht the net grvittionl potentil energy of prticle when t the origin is U(0,0) 4 G /. Hint: The eqution bove gives the grvittionl potentil energy of ss 1 in the presence of ss, or vice vers. Use it nd superposition to find the net potentil energy of ss in the presence of the four fied sses. As long s the sses reined fied, it is not necessry to consider their utul potentil energies. y (b) If the prticle is relesed t rest fro infinity nd psses through the origin, clculte the gnitude of its velocity there.
(c) Clculte the net potentil energy of when it is t the id-point of the right-hnd side of the squre. y (d) If the prticle is relesed t rest fro the id-point of the right-hnd side of the squre, does it rech the origin? Support your nswers with physicl rguents. (e) In Discussion Question B, you clculted the force on the ss when it is t the id-point of the right-hnd side of the squre. If ll went well, you found tht this force points in the direction, i.e. towrds the origin. Given this fct, plus your nswer to prt (d), cn you ke rough sketch of the potentil energy function U() for points long the -is? Give it try! U y / / 3