Physcs 207 Lecture 6 Agenda: Physcs 207, Lecture 6, Sept. 25 Chapter 4 Frames of reference Chapter 5 ewton s Law Mass Inerta s (contact and non-contact) Frcton (a external force that opposes moton) Free Body Dagrams (a very mportant tool) Assgnment: For Wednesday read Chapter 6 WebAssgn Problem Set 2 due Wednesday noon WebAssgn Problem Set 3 avalable today MdTerm Thursday, Oct. 5, Chapters 1-6, 1 90 mnutes, 7-8:457 PM Physcs 207: Lecture 6, Pg 1 Relatve moton and frames of reference Reference frame S s statonary Reference frame S s movng at v o Ths also means that S moves at v o relatve to S Defne tme t = 0 as that tme when the orgns concde Physcs 207: Lecture 6, Pg 2 Relatve Velocty Two observers movng relatve to each other generally do not agree on the outcome of an experment (path) For example, observers A and B below see dfferent paths for the ball Relatve Velocty, r, v, a and r,, v v, a a The postons as seen from the two reference frames are related through the velocty (remember S s movng at a constant v 0 relatve to S ) r = r v o t The dervatve of the poston equaton wll gve the velocty equaton v = v v o = d (r v o t)/dt Physcs 207: Lecture 6, Pg 3 Physcs 207: Lecture 6, Pg 4 Acceleraton n Dfferent Frames of Reference The dervatve of the velocty equaton wll gve the acceleraton equaton v = v v o a = a The acceleraton of the partcle measured by an observer n one frame of reference s the same as that measured by any other observer movng at a constant velocty relatve to the frst frame. Monkey and Hunter A hunter sees a monkey n a tree, ams hs gun at the monkey and fres. At the same nstant the monkey lets go. Does the bullet (now the movng frame): A Go over the monkey B Ht the monkey C Go under the monkey End of Chapter 5 Physcs 207: Lecture 6, Pg 5 Physcs 207: Lecture 6, Pg 6 Page 1
Physcs 207 Lecture 6 Chapter 6: ewton s Laws and s Sr Issac ewton (1642-1727) Dynamcs See text: Chapter 5 Prncpa Mathematca publshed n 1687. Ths revolutonary work proposed three laws of moton: Law 1: An obect subect to no external forces s at rest or moves wth a constant velocty f vewed from an nertal reference frame. Law 2: For any obect, F ET = Σ F = ma Law 3: s occur n pars: F A, B = - F B, A (For every acton there s an equal and opposte reacton.) Read: of B on A Physcs 207: Lecture 6, Pg 7 Physcs 207: Lecture 6, Pg 8 We have a general noton of forces s from everyday lfe. In physcs the defnton must be precse. A force s an acton whch causes a body to accelerate. (ewton s Second Law) Examples: Contact s (physcal contact between obects) Kckng a ball Feld s (on-contact) (acton at a dstance) Moon and Earth See text: 5-1 On a mcroscopc level, all forces are non-contact Mass See text: 5-3 We have an dea of what mass s from everyday lfe. In physcs: mass (n Phys 207) s a quantty that specfes how much nerta an obect has (.e. a scalar that relates force to acceleraton) (ewton s Frst Law) Mass s an nherent property of an obect. Mass and weght are dfferent quanttes; weght s usually the magntude of a gravtatonal (non-contact) force. Pound (lb) s a defnton of weght (.e., a force), not a mass! Physcs 207: Lecture 6, Pg 9 Physcs 207: Lecture 6, Pg 10 See text: 5-2 Inerta and Mass The tendency of an obect to resst any attempt to change ts velocty s called Inerta Mass s that property of an obect that specfes how much resstance an obect exhbts to changes n ts velocty Mass s an nherent property of an obect Mass s ndependent of the obect s surroundngs Mass s ndependent of the method used to measure t Mass s a scalar quantty The SI unt of mass s kg ewton s Frst Law and IRFs An obect subect to no external forces moves wth a constant velocty f vewed from an nertal reference frame (IRF). If no net force actng on an obect, there s no acceleraton. The above statement can be used to defne nertal reference frames. An IRF s a reference frame that s not acceleratng (or rotatng) wth respect to the fxed stars. If one IRF exsts, nfntely many exst snce they are related by any arbtrary constant velocty vector! The surface of the Earth may be vewed as an IRF Physcs 207: Lecture 6, Pg 11 Physcs 207: Lecture 6, Pg 12 Page 2
Physcs 207 Lecture 6 ewton s Second Law See text: 5-4 The acceleraton of an obect s drectly proportonal to the net force actng upon t. The constant of proportonalty s the mass. Lecture 6, Exercse 1 ewton s Second Law A constant force s exerted on a cart that s ntally at rest on an ar table. The force acts for a short perod of tme and gves the cart a certan fnal speed. Cart Ar Track Ths expresson s vector expresson: F x, F y, F z Unts The metrc unt of force s kg m/s 2 = ewtons () The Englsh unt of force s Pounds (lb) For a second shot, we apply a force only half as large. To reach the same fnal speed, how long must the same force force be appled? (A) 4 x as long (B) 2 x as long (C) 1/2 as long (D) 1/4 as long Physcs 207: Lecture 6, Pg 13 Physcs 207: Lecture 6, Pg 14 Lecture 6, 6 Exercse 1 ewton s Second Law Soluton Cart Ar Track F = ma Snce F 2 = 1/2 F 1 a 2 = 1/2 a 1 We know that under constant acceleraton, v = a t So, a 2 t 2 = a 1 t 1 we want equal fnal veloctes 1/2 a 1 / t 2 = a 1 / t 1 t2 = 2 t 1 (B) 2 x as long Lecture 6, Exercse 2 ewton s Second Law A force of 2 ewtons acts on a cart that s ntally at rest on an ar table wth no ar and pushed for 1 second. Because there s no ar, the cart stops mmedately after I fnsh pushng. It has traveled a dstance, D. Cart Ar Track ext, the force of 2 ewtons acts agan but s appled for 2 seconds. The new dstance the cart moves relatve to D s: (A) 8 x as far (B) 4 x as far (C) 2 x as far (D) 1/4 as far Physcs 207: Lecture 6, Pg 15 Physcs 207: Lecture 6, Pg 16 Lecture 6, 6 Exercse 2 Soluton Cart Ar Track ewton s Thrd Law: See text: 5-6 If obect 1 exerts a force on obect 2 (F 2,1 ) then obect 2 exerts an equal and opposte force on obect 1 (F 1,2 ) We know that under constant acceleraton, x = a ( t) 2 /2 (when v 0 =0) Here t 2 =2 t 1, F 2 = F 1 a 2 = a 1 1 2 ( 2 ) 4 1 2 2 x a t2 2 t = = 1 = x 2 2 1 a t t 1 1 2 F 1,2 = -F 2,1 For every acton there s an equal and opposte reacton IMPORTAT: ewton s 3 rd law concerns force pars whch act on two dfferent obects (not on the same obect)! (B) 4 x as long Physcs 207: Lecture 6, Pg 17 Physcs 207: Lecture 6, Pg 18 Page 3
Physcs 207 Lecture 6 Two Examples (non-contact) Consder the forces on an obect undergong proectle moton Lecture 6, Exercse 3 ewton s Thrd Law A fly s deformed by httng the wndsheld of a speedng bus. EARTH F B,E = - m B g F E,B = m B g F B,E = - m B g F E,B = m B g The force exerted by the bus on the fly s, (A) greater than (B) the same as (C) less than that exerted by the fly on the bus. Physcs 207: Lecture 6, Pg 19 Physcs 207: Lecture 6, Pg 20 Lecture 6, Exercse 4 ewton s Thrd Law Same scenaro but now we examne the acceleratons Lecture 6, Exercses 3&4 ewton s Thrd Law Soluton By ewton s thrd law these two forces form an nteracton par whch are equal (but n opposng drectons). The magntude of the acceleraton, due to ths collson, of the bus (A) greater than (B) the same as (C) less than that of the fly. Physcs 207: Lecture 6, Pg 21 Thus the forces are the same However, by ewton s second law F net = ma or a = F net /m. So F b, f = -F f, b = F 0 but a bus = F 0 / m bus << a fly = F 0 /m fly Answer for acceleraton s (C) Physcs 207: Lecture 6, Pg 22 Free Body Dagram A heavy sgn s hung between two poles by a rope at each corner extendng to the poles. Free Body Dagram T 1 T 2 1 2 Eat at Bob s Eat at Bob s 1 T 1 T 2 2 What are the forces on the sgn? Add vectors Physcs 207: Lecture 6, Pg 23 Physcs 207: Lecture 6, Pg 24 Page 4
Physcs 207 Lecture 6 Free Body Dagram T 1 T 2 1 2 Eat at Bob s ormal s Certan forces act to keep an obect n place. These have what ever force needed to balance all others (untl a breakng pont). F B,T Vertcal : y-drecton 0 = - + T 1 sn 1 + T 2 sn 2 Horzontal : x-drecton 0 = -T 1 cos 1 + T 2 cos 2 F T,B Physcs 207: Lecture 6, Pg 25 Physcs 207: Lecture 6, Pg 26 Pars ewton s 3rd law concerns force pars: Two members of a force par cannot act on the same obect. Don t mx gravtatonal (a non-contact force of the Earth on an obect) and normal forces. They must be vewed as separate force pars (consstent wth ewton s 3 rd Law) F B,E = - F B,T F T,B Lecture 6, Exercse 5 ewton s 3rd Law Two blocks are beng pushed by a fnger on a horzontal frctonless floor. How many acton-reacton force pars are present n ths exercse? a (A) 2 (B) 4 (C) 6 b F E,B = Physcs 207: Lecture 6, Pg 27 Physcs 207: Lecture 6, Pg 28 Lecture 6, Exercse 5 Soluton: F a,f F f,a F b,a F a,b F g,a a F E,a F g,b b F E,b Example Consder the followng two cases (a fallng ball and ball on table), Compare and contrast Free Body Dagram and Acton-Reacton Par sketch F a,g F b,g 246 F a,e F b,e Physcs 207: Lecture 6, Pg 29 Physcs 207: Lecture 6, Pg 30 Page 5
Physcs 207 Lecture 6 Example The Free Body Dagram Example Frst: Free-body dagram Second: Acton/reacton par forces F B,T = F B,E = - F B,T = F T,B = - F B,E = - Ball Falls For Statc Stuaton = F E,B = F E,B = Physcs 207: Lecture 6, Pg 31 Physcs 207: Lecture 6, Pg 32 See text: Example 5.7 Exercse: Frctonless nclned plane A block of mass m sldes down a frctonless ramp that makes angle wth respect to horzontal. What s ts acceleraton a? Frctonless nclned plane... See text: Example 5.7 Defne convenent axes parallel and perpendcular to plane: Acceleraton a s n x drecton only (defned as a x ). m a m a Physcs 207: Lecture 6, Pg 33 Physcs 207: Lecture 6, Pg 34 Frctonless nclned plane... See text: Example 5.7 Angles of the nclned plane See text: Example 5.7 Use a FBD and consder x and y components separately: F x : ma x = sn a x = g sn F y : ma y = 0 = cos = cos ma x = sn ma x sn cos + φ = 90 φ Physcs 207: Lecture 6, Pg 35 Physcs 207: Lecture 6, Pg 36 Page 6
Physcs 207 Lecture 6 See text: 5.8 See text: 5.8 A specal contact force, frcton What does t do? It opposes moton! Frcton... Frcton s caused by the mcroscopc nteractons between the two surfaces: How do we characterze ths n terms we have learned? Frcton results n a force n a drecton opposte to the drecton of moton (actual or, f statc, then mpled)! F APPLIED ma f FRICTIO Physcs 207: Lecture 6, Pg 37 Physcs 207: Lecture 6, Pg 38 See text: 5.8 See text: 6-1 Frcton... Model for Sldng Frcton (wth moton) of frcton acts to oppose moton: Parallel to a surface Perpendcular to a ormal force. The drecton of the frctonal force vector s perpendcular to the normal force vector. The magntude of the frctonal force vector f K s proportonal to the magntude of the normal force. f K = µ K ( = µ K n the prevous example) F ma The heaver somethng s, the greater the frctonal force f F The constant µ K s called the coeffcent of knetc frcton. See fgure 5.17 Physcs 207: Lecture 6, Pg 39 Physcs 207: Lecture 6, Pg 40 Case study... See text: 6-1 Dynamcs: x-axs : ma x = F µ K y-axs : ma y = 0 = or = Statc Frcton... See text: Ch 5.8 So far we have consdered frcton actng when somethng has a non-zero velocty We also know that t acts n fxed or statc systems: so F µ K = m a x f k v In these cases, the force provded by frcton depends on the forces appled on the system (f s µ ) Opposes moton that would occur f µ were zero f k F ma x F net µ K f S Physcs 207: Lecture 6, Pg 41 Physcs 207: Lecture 6, Pg 42 Page 7
Physcs 207 Lecture 6 See text: Ch 5.8 See text: Ch 5.8 Statc Frcton... Just lke n the sldng case except a = 0. : F net f S = 0 : = Whle the block s statc: f S = F net (unlke knetc frcton) Statc Frcton... The maxmum possble force that the frcton between two obects can provde s f MAX = µ S, where µ s s the coeffcent of statc frcton. So f S µ S. As one ncreases F, f S gets bgger untl f S = µ S and the obect breaks loose and starts to move. F net F f S f S Physcs 207: Lecture 6, Pg 43 Physcs 207: Lecture 6, Pg 44 See text: Ch 5.8 See text: 6-1 Statc Frcton... µ S s dscovered by ncreasng F untl the block starts to slde: : F MAX µ S = 0 : = µ S = F MAX / Actve Fgure Addtonal comments on Frcton: Snce f = µ, the force of frcton does not depend on the area of the surfaces n contact. Logc dctates that µ S > µ K for any system F MAX µ S Physcs 207: Lecture 6, Pg 45 Physcs 207: Lecture 6, Pg 46 Sept. 25 Recappng Chapter 4 Frames of reference Chapter 5 ewton s Law Mass Inerta s (contact and non-contact) Frcton (a external force that opposes moton) Free Body Dagrams (a very mportant tool) Assgnment: For Wednesday read Chapter 6 WebAssgn Problem Set 2 due Wednesday noon WebAssgn Problem Set 3 avalable today MdTerm Thursday, Oct. 5, Chapters 1-6, 1 90 mnutes, 7-8:457 PM Physcs 207: Lecture 6, Pg 47 Page 8