PHY 140Y FOUNDATIONS OF PHYSICS Problem Set #2

Transcription

1 PHY 140Y FOUNDATIONS OF PHYSICS Prble Set #2 HANDED OUT: DUE: Friday, Octber 5, 2001 (in cla) 5:00 PM, Thurday, Octber 18, 2001 in the apprpriate bx, labeled by tutrial grup, in the baeent at the btt f the tair leading dwn fr MP202 LATE PENALTY: 5 ark/day (which al applie t weekend day!) until 1:00 PM, Mnday, Octber 22, after which it will nt be accepted a lutin will then be available in tutrial and n the WWW NOTES: Anwer all quetin A elected ubet (3-4) will be arked ut f 100% Mark will be given fr wrking and unit, a well a fr final anwer QUESTIONS: 1 A pace tatin 120 in diaeter i et rtating in rder t give it ccupant artificial gravity Over a perid f 50 inute, all rcket bring the tatin teadily t it final rtatin rate f 1 revlutin every 20 ecnd What are the radial and tangential acceleratin f a pint n the ri f the tatin 20 inute after the rcket tart firing? 2 Yu ay have nticed that a a caette tape i played, the radiu f the tape left n the upply pl decreae lwly with tie a the tape i drawn thrugh the achine at a cntant peed v Becaue the radiu f the tape reaining change lwly cpared with the peed f the tape thrugh the achine, t a gd apprxiatin we can write: d v r rω > 0 dt where ω i the angular peed f the tape reel Ntice that the tape peed v i cntant, but radiu r change lwly with tie Therefre the angular peed f the reel al change lwly with tie Aue that r repreent the average radiu f each ciled layer (ince the tape i actually wund in a piral) Let r be the initial radiu f the tape when t 0 and 0 rad, where i the ttal angle f tape unwund fr the pl Repreent the thickne f the tape by b When the tape ha unwund thrugh a ttal angle, the radiu f tape reaining i r r b dr b d b v (a) Shw that dt 2π dt 2π r (b) Integrate the reult f part (a) t hw that PHY 140Y Fundatin f Phyic (K Strng) Prble Set 2, page 1 2π 2 2 π r r ) bv t ( What i the geetrical interpretatin f the left-hand ide f thi equatin? What i the phyical interpretatin f the right-hand ide f thi equatin? Illutrate each with a ketch

2 (c) Slve the reult f part (b) fr r and ubtitute thi fr r in Equatin (1) Integrate the reulting exprein t hw that the ttal angle thrugh which the tape ha turned at tie t π i 2 2 bv t r r b π a Hint: The fllwing integratin frula ay be ueful: dt 2 b at + C b at 3 A kayaker attept t cr a river in which the current ha a unifr peed f 50 k/hr acr the 500 width f the river The kayaker i paddling at the frenetic peed f 100 k/hr with repect t the water and want t travel traight acr the river t avid a huge waterfall jut dwntrea (a) What i the reulting peed f the kayak with repect t the grund? (b) At what angle t the trea flw huld the kayaker paddle that the tranit i ade withut ving uptrea r dwntrea? (c) Hw lng de the trip take? 4 Iagine nw falling vertically dwnward at cntant peed v A car i driving hrizntally in a traight line thrugh the nw at cntant peed v (a) Shw that accrding t the driver, the nw appear t fall alng a directin that ake an angle with the vertical directin, where tan v / v Include a ketch f the velcity vectr If yu tilt an ubrella at thi angle while walking at peed v in nw (r rain) falling vertically at peed v, yu keep driet (b) Thi prble al ha iplicatin in atrny Iagine the nw t be tarlight ving at cntant peed c tward the lar yte, a hwn in the figure belw The Earth, in it rbital tin, travel at peed v Oberver n the Earth ee the light cing fr a directin that i nt in the directin f the true pitin f the tar The angle i fund fr tan v / c Thu a telecpe ut be tilted in the directin f the rbital velcity f the Earth by an angle t ee the tar Over the cure f a year, the tar execute a all circle f angular radiu abut it true pitin perpendicular t the plane f the rbit f the Earth Thi effect i knwn a tellar aberratin The rbital peed f the Earth i abut 30 k/ Evaluate the angle If the tar i in the plane f the rbit, then the aberratinal path f the tar i a line centered n the true pitin f the tar If the tar i between the plane f the rbit and the perpendicular t the rbit, then the aberratinal path f the tar i an ellipe light fr a ditant tar c v v Earth in apprxiately circular rbit, in perpective but nt t cale PHY 140Y Fundatin f Phyic (K Strng) Prble Set 2, page 2

3 5 When the tarting gun fire, a 600 kg printer accelerate with a cntant acceleratin fr the firt 100 f the track and then run with cntant peed fr the duratin f the race, cpleting the 100 print in 100 What i the agnitude f the ttal frce cauing the acceleratin f the printer? 6 A car travelling at cntant peed reache the cret f a huge hill that i a cylindrical cr ectin f radiu 70 a hwn in the figure belw (a) At the intant when the car i at the cret f the hill, ake a Secnd Law frce diagra indicating cheatically all frce acting n the car (b) What i the axiu peed that the car can travel withut becing airbrne at the cret f the hill? Expre yur reult in k/hr R 70 7 A large Ferri wheel f radiu 150 i rtating wildly ut f cntrl at 400 rev/in A tudent f a 800 kg i becing increaingly uncfrtable with the ride On the ther hand, yu (f a 500 kg) are rather enjying the wild ride T cal yur cpanin panic, yu decide t explain thingly the phyic f the effect that yu are bth experiencing n the Ferri wheel Yu firt pint ut that during the entire ride, there are nly tw frce acting n each f yu: (1) yur weight, and (2) the nral frce f the bench eat n yur backide (a) When the chair i at the tp f the circle, which f the tw frce ha the greater agnitude? What i the agnitude f the difference between yur weight and the nral frce f the eat n yur backide? (b) When the chair i at the btt f the circle, which f the tw frce ha the greater agnitude? What i the difference in the agnitude f the frce n yu at thi lcatin? (c) When the chair i nt at the tp r btt f the circle, the tw frce are n lnger antiparallel t each ther, ince the chair wivel At what tw place in the tin i the deviatin f the directin f the nral frce fr the lcal vertical a axiu? At thee lcatin, what i the angle between the directin f thi frce and the lcal vertical directin? (d) At what angular peed ut the wheel rtate that yu are bth apparently weightle at the tp f the pin? Give the reult in rev/in Are yu truly weightle at the tp f the pin? Explain why r why nt Hint: Yu ay need t ue the Law f Cine Al, an extreu f functin f(x) can be fund uing df(x)/dx0 PHY 140Y Fundatin f Phyic (K Strng) Prble Set 2, page 3

4 8 A tring ade fr braided dental fl ha a axiu periible tenin f 500 N and i ued by a 600 kg inate t ecape prin (apparently thi actually happened!) The inate tart at ret and lide vertically dwn the tring fr 80 (a) Draw a frce diagra indicating all frce acting n the inate in the decent (b) Deterine the agnitude f the acceleratin f the inate that the tring barely de nt break I thi a iniu r a axiu value? Explain (c) What i the axiu peed f the frer inate at the end f the 80 decent if the tring de nt break? 9 Aueent park ccainally have a large cylindrical tube unted t pin abut the vertical axi f the cylinder a hwn in the figure belw Patrn tand againt the wall f the cylinder and it i et int rtatin At a certain critical angular velcity, the flr n which the patrn are tanding i lwered and the patrn are left hanging n the wall (but n the verge f lipping) due t the frce f tatic frictin n the Let R be the radiu f the cylinder, the a f ne patrn, and the cefficient f tatic frictin fr thi patrn n the wall (a) Draw the frce diagra fr the patrn, indicating all frce acting n the patrn (b) I there an acceleratin f the patrn in the vertical directin? What de thi iply abut the relatinhip between the tatic frictinle frce n the patrn and the weight f the patrn? Are thee frce a Newtn Third Law pair? (c) I there an acceleratin f the patrn in the hrizntal directin? Hw i thi acceleratin related t the peed f the patrn? (d) Hw i the angular velcity ω f the cylinder related t the tangential peed f the patrn? If the patrn i ready t lip, find the angular velcity ω f the cylinder in ter f R,, and g Expre yur anwer in rev/ (e) If the flr f the cylinder drp a ditance L, hw lng de it take fr the patrn t lide dwn the ditance L? Cnider the cefficient f kinetic frictin t be k ω ω R L PHY 140Y Fundatin f Phyic (K Strng) Prble Set 2, page 4

5 10 Tw ae, 1 and 2, are cnnected by an ideal crd a hwn in the figure belw The crd pae ver an ideal pulley A tudent deterine that when the urface i inclined at an angle t the hrizntal, 1 i jut n the verge f lipping t the right up the incline (a) Draw a frce diagra hwing all frce acting n a 2 (b) Draw a frce diagra hwing all frce acting n a 1 (c) Intrduce apprpriate crdinate yte() and hw that the cefficient f tatic frictin fr 1 n the inclined plane i 2 1 in 1 c Nw the inclinatin angle f the inclined plane i increaed t angle φ that the a 1 i jut n the verge f lipping t the left dwn the incline (d) Draw a frce diagra hwing all frce acting n a 2 (e) Draw a frce diagra hwing all frce acting n a 1 (f) Shw that the cefficient f tatic frictin fr 1 n the inclined plane i nw 1 in φ 2 1 cφ Ntice that if the upended a 2 0 kg, then φ 0 and thi equatin reduce t tan φ tan (g) I thi the ae cefficient f frictin a that deterined in part (c)? Why r why nt? (h) Shw that the rati f the tw ae i given by the rather elegant exprein 2 tan + tan φ ec + ecφ 1 1 φ 2 PHY 140Y Fundatin f Phyic (K Strng) Prble Set 2, page 5