10/24/2013. PHY 113 C General Physics I 11 AM 12:15 PM TR Olin 101. Plan for Lecture 17: Review of Chapters 9-13, 15-16

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1 0/4/03 PHY 3 C General Physcs I AM :5 PM T Oln 0 Plan or Lecture 7: evew o Chapters 9-3, 5-6. Comment on exam and advce or preparaton. evew 3. Example problems 0/4/03 PHY 3 C Fall Lecture 7 0/4/03 PHY 3 C Fall Lecture 7 Webassgn questons Assgnment #5 Consder the snusodal wave o the gure below wth the wave uncton y = 0.50 cos(5.7x 50.3t) where x and y are n meters and t s n seconds. At a certan nstant, let pont A be at the orgn and pont B be the closest pont to A along the x axs where the wave s 43.0 out o phase wth A. What s the coordnate o B? o x o 80 0/4/03 PHY 3 C Fall Lecture 7 3

2 0/4/03 Webassgn questons Assgnment #5 A transverse wave on a strng s descrbed by the ollowng wave uncton. y = 0.5 sn ((π/9)x + 5πt) where x and y are n meters and t s n seconds. (a) Determne the transverse speed at t = 0.50 s or an element o the strng located at x =.50 m. y( x, t) 0.5 cos x 5t t 9 9 (b) Determne the transverse acceleraton at t = 0.50 s or an element o the strng located at x =.50 m. y( x, t) 0.5 sn x 5t t 9 9 0/4/03 PHY 3 C Fall Lecture 7 4 Webassgn questons Assgnment #5 A snusodal wave n a rope s descrbed by the wave uncton y = 0.0 sn (0.69πx + 0πt) y0 snkx t where x and y are n meters and t s n seconds. The rope has a lnear mass densty o 0.30 kg/m. The tenson n the rope s provded by an arrangement lke the one llustrated n the gure below. What s the mass o the suspended object? T c k T mg mg 0/4/03 PHY 3 C Fall Lecture 7 5 Comment about exam on Tuesday 0/9/03 0/4/03 PHY 3 C Fall Lecture 7 6

3 0/4/03 clcker queston What s the purpose o exams? A. Pure pan and suerng or all nvolved. B. To measure what has been learned. C. To help students learn the materal. D. Other. 0/4/03 PHY 3 C Fall Lecture 7 7 Advce on how to prepare or the exam evew lecture notes and text chapters 9-3,5-6 Prepare equaton sheet Work practce problems Topcs covered Lnear momentum otatonal moton and angular momentum Gravtatonal orce and crcular orbts Statc equlbrum Smple harmonc moton Wave moton 0/4/03 PHY 3 C Fall Lecture 7 8 What to brng to exam: Clear head Calculator Equaton sheet Pencl or pen 0/4/03 PHY 3 C Fall Lecture 7 9 3

4 0/4/03 clcker queston: Have you looked at last year s exams? A. Yes B. No 0/4/03 PHY 3 C Fall Lecture 7 0 Lnear momentum What s t? When s t conserved? Conservaton o momentum n analyss o collsons Noton o center o mass Dene "lnear momentum": mv p Unts o lnear momentum : kg m/s N s dv dmv dp F ma m F dp Impulse: I F dp p p 3 0/4/03 PHY C Fall Lecture 7 Lnear momentum -- contnued Physcs o composte systems Newton' s second law : dv F ma m Note that d 0/4/03 PHY 3 C Fall Lecture 7 F 0, then : p 0 p (constant) p ntal p nal d m v d p 4

5 0/4/03 Example completely nelastc collson; balls movng n one dmenson on a rctonless surace m v m v v For p ntal v p 0/4/03 PHY 3 C Fall Lecture 7 3 mv mv m m m 0.3kg, m 0.5kg v m / sˆ, v m / sˆ nal m m 0.3 ˆ 0.5 ˆ m/s ˆ v m / s Examples o two-dmensonal collson; balls movng on a rctonless surace p ntal p nal mv mv cos mv cos 0 mv sn mv sn Knowns : m, m, v Unknowns : v, v,, Need more equatons - - 0/4/03 PHY 3 C Fall Lecture 7 4 The noton o the center o mass and the physcs o composte systems Newton' s second law : Dene : F F F m a r total M d r M d r m m r M d m r m 0/4/03 PHY 3 C Fall Lecture 7 5 5

6 0/4/03 Fndng the center o mass r m r M M m In ths example : m m kg; m3 kg m ˆ ˆ ˆ x mx m3 y3j r m m m3 ()(m )ˆ m ˆ mˆ j r mˆ.00mˆj 0/4/03 PHY 3 C Fall Lecture 7 6 otatonal moton and angular momentum Angular varables Newton s law or angular moton otatonal energy Moment o nerta Angular momentum d d 0/4/03 PHY 3 C Fall Lecture 7 7 evew o rotatonal energy assocated wth a rgd body otatonal energy : K rot m v m r where I m r m r I 0/4/03 PHY 3 C Fall Lecture 7 8 6

7 0/4/03 Moment o nerta: I m r I Ma I Ma mb 0/4/03 PHY 3 C Fall Lecture 7 9 Total knetc energy o rollng object : K total K rollng K I Mv Note that : d ds d v K total K rollng K I Mv I M v 0/4/03 PHY 3 C Fall Lecture 7 0 clcker exercse: Three round balls, each havng a mass M and radus, start rom rest at the top o the nclne. Ater they are released, they roll wthout slppng down the nclne. Whch ball wll reach the bottom rst? I A M B C I B M 0.5M A I C M 0.4M 5 K U K U I 0 Mgh M M gh v I / M 0/4/03 PHY 3 C Fall Lecture 7 v 0 7

8 0/4/03 How can you make objects rotate? Dene torque: t = r x F t = rf sn F ma r F τ r ma Iα r sn r F Newton's law or rotatonal moton : τ Iα 0/4/03 PHY 3 C Fall Lecture 7 Example orm Webassgn # X t 3 clcker exercse When the pvot pont s O, whch torque s zero? A. t? B. t? C. t 3? t t 0/4/03 PHY 3 C Fall Lecture 7 3 Vector cross product; rght hand rule C AB C A B sn ˆ ˆ ˆj ˆj kˆ kˆ 0 ˆ ˆj ˆj ˆ kˆ ˆj kˆ kˆ ˆj ˆ kˆ ˆ ˆ kˆ ˆj 0/4/03 PHY 3 C Fall Lecture 7 4 8

9 0/4/03 From Newton s second law contnued conservaton o angular momentum: d r F τ r p Dene : L r p dl I τ 0 0 L (constant) 0/4/03 PHY 3 C Fall Lecture 7 5 Example o conservaton o angular momentum L L b b wheel L L wheel b L L b 0 L L L wheel wheel wheel 0/4/03 PHY 3 C Fall Lecture 7 6 Summary conservaton laws we have studed so ar Conserved quantty Necessary conon Lnear momentum p F net = 0 Angular momentum L t net = 0 Mechancal energy E No dsspatve orces 0/4/03 PHY 3 C Fall Lecture 7 7 9

10 0/4/03 Fundamental gravtatonal orce law and planetary moton Newton s gravtatonal orce law Gravty at Earth s surace Crcular orbts o gravtatonal bodes Energy assocated wth gravtaton and orbtal moton 0/4/03 PHY 3 C Fall Lecture 7 8 Unversal law o gravtaton Newton (wth help rom Galleo, Kepler, etc.) 687 Gm m rˆ F r G N m kg 0/4/03 PHY 3 C Fall Lecture 7 9 Gravtatonal orce o the Earth E m GM m F E E GM g E E (6.370 ) 4 m/s 9.8m/s Note: Earth s gravty acts as a pont mass located at the Earth s center. 0/4/03 PHY 3 C Fall Lecture

11 0/4/03 Stable crcular orbt o two gravtatonally attracted objects (such as the moon and the Earth) EM F a v Newton's law or Moon due to Earth : F M a M v a EM v ω M EM M GM T π GM π T 3 EM E EM E EM 8 3 (3.840 ) π s 7.4 days 0/4/03 PHY 3 C Fall Lecture 7 3 Crcular orbtal moton about center o mass m v Note that v m T T G m m m then v m Gmm m m T T 3 3 m Gm v m m T m m T 3 G m m v 0/4/03 PHY 3 C Fall Lecture 7 3 dl τ 0 L (const) v m L =m v L =m v L = L + L m v L L Note: More generally, stable orbts can be ellptcal. 0/4/03 PHY 3 C Fall Lecture 7 33

12 0/4/03 Gravtatonal potental energy Example: U U gravty gravty ( r) ( r) r r rre F dr Gmm r' Gmm dr' r Gmmrˆ F r U gravty GM EmS ( r E h) h E 0/4/03 PHY 3 C Fall Lecture 7 34 Analyss o statc equlbrum Meanwhle back on the surace o the Earth: Conons or stable equlbrum Balance o orce : Balance o torque : F 0 τ 0 0/4/03 PHY 3 C Fall Lecture 7 35 Torques : F (m ) mg ( ) g 0 0/4/03 PHY 3 C Fall Lecture 7 36

13 0/4/03 t 0 T mgx Mg T sn 0 mgx / Mg / T sn For 53 mg 600 N T 33 N o 8m x m Mg 00 N * *X x mg / Mg 0/4/03 PHY 3 C Fall Lecture 7 37 Some practce problems 0/4/03 PHY 3 C Fall Lecture /4/03 PHY 3 C Fall Lecture

14 0/4/03 From webassgn: A 00-kg merry-go-round n the shape o a unorm, sold, horzontal dsk o radus.50 m s set n moton by wrappng a rope about the rm o the dsk and pullng on the rope. What constant orce would have to be exerted on the rope to brng the merry-go-round rom rest to an angular speed o rev/s n.00 s? (State the magntude o the orce.) vew rom top: F τ r F Iα I M 0/4/03 PHY 3 C Fall Lecture 7 40 From webassgn: A 0.3-kg monkey clmbs a unorm ladder wth weght w =.4 0 N and length L = 3.35 m as shown n the gure below. The ladder rests aganst the wall and makes an angle o θ = 60.0 wth the ground. The upper and lower ends o the ladder rest on rctonless suraces. The lower end s connected to the wall by a horzontal rope that s rayed and can support a maxmum tenson o only 80.0 N. 0/4/03 PHY 3 C Fall Lecture 7 4 4