Chter 4 orce nd Neton s Ls of Motion e cn describe motion, but h do things move? orces: Objects interct through forces. A force is ush or ull. orces cn be long rnge (grvit, electric, mgnetic, etc.) or contct (norml force, tension, etc.). ig. 04.01 Obviousl, forces re vector quntities since their effect deends on the directions of the forces. The net force is the vector sum of ll forces cting on n object. net 1 n A free-bod digrm (BD) is n essentil tool for finding the net force cting on n object. (See ge 91.) Dr the object in simlified Identif ll the forces tht re eerted on the object. Dr vector rros reresenting ll the forces on the object. 4-1
Emles 1. reel flling object.. Object hnging from roe. 3. Object sitting on horizontl tble. 4. Object sitting on horizontl tble being ulled b roe. REE BODY DIAGRAMS GO HERE. I ill dr them on the bord. Dring the free-bod digrm is the ke to solving roblems. Neton s irst L (l of inerti): An object s velocit vector v remins constnt if nd onl if the net force cting on the object is zero. An object moving t constnt velocit hs no net force! A revolutionr ide. An object moving t constnt velocit is sid to be in trnsltionl equilibrium. Tht velocit could be zero. Inerti is the resistnce to chnges in velocit. Neton s Second l: The rte of chnge of n object s velocit is roortionl to the net force cting on it nd inversel roortionl to its mss. m Recll our rule: e never del ith vectors, e del ith their comonents. A fr more useful form of Neton s second l ill be m m 4-
The left hnd side is sulied b the free-bod digrm. The right hnd side is sulied b our knoledge of the motion. The SI unit of force is the neton. 1 N = 1 kg m/s. ht is mss? Mss is mesure of inerti. Mss is not the sme s eight. Neton s Third L: In n interction beteen to objects, ech object eerts force on the other. These forces re equl in mgnitude nd oosite in direction. If to objects A nd B re eerting forces on ech other, BA A B AB AB BA The forces re equl in mgnitude nd oosite in direction. Neton s Ls of Motion (ges 9-99) 1. An object s velocit vector v remins constnt if nd onl if the net force cting on the object is zero.. hen nonzero net force cts on n object, the object s velocit chnges. The object s ccelertion if roortionl to the net force cting on it nd inversel roortionl to its mss. m m 3. In n interction beteen to objects, ech object eerts force on the other. These forces re equl in mgnitude nd oosite in direction. 4-3
AB BA Interction Pirs A simle emle: I ush don on the tble ith force of 15 ounds. The tble ushes u on me ith force of 15 ounds. A little hrder emle. A block sits t rest on flt surfce. The block is held don b its eight. ht is the interction ir for the eight? N Anser: It is not the norml force! The eight is the force of the erth ulling don on the object. The interction ir is the object ulls u on the erth. To find the rection to the ction, sitch nouns nd directions. The third l rdo. If I eert 15 ound force on chir, b Neton s third l, the chir ill ull bck ith 15 ound force. The forces re equl in mgnitude nd oosite in direction, so the cncel. I cnnot move the chir. Obviousl, I cn move the chir. h? The 15 ound forces ct on different sstems. The net force on the chir is the difference beteen m 15 ound force nd the friction cting on the chir s it moves cross the floor. As long s I cn ush hrder thn friction, the chir ill ccelerte in the direction I m ushing. Grvittionl orces e lred rgued the mgnitude of the eight cn be ritten s mg h do things hve eight? The erth ulls don on them. h? No one rell knos. Ho does it ull don? The erth is not touching the bll et it still ulls don. Ho does the erth ull it don? No one rell knos. Generl reltivit elins some sects of grvit, but the interction mechnism is unelined. 4-4
Neton ostulted tht to mss m1 nd m re ttrcted to ech other b force given b Gm1m r 11 here r is the distnce beteen the to msses nd G 6.6710 Nm /kg is the universl grvittionl constnt. Since G is so smll, it tkes lot of mss ( lnet for instnce) to eerience sizble grvittionl force. Since the grvittionl force cuses n object to hve eight, it cn be shon (see ges 100-101) tht GM g R E E ME is the mss of the erth nd RE its rdius. Henr Cvendish mesured the vlue of G: htt://.outube.com/tch?v=pdiuok9n The distnce r is mesure from the center of the lnet nd not its surfce. Contct orces Norml force N is the force of surfce ushing u on n object. It is force of constrint tht forces the object to onl move long the surfce. Accordingl it is erendiculr to the surfce 4-5
(hence its nme, norml force). In the first cse, the norml force is equl in mgnitude to the eight. But in the other to cses, the re not equl. riction force f is the force of surfce oosing the motion of the object. The surfce holds onto the object nd resists its motion. f v The norml force nd the friction force re comonents of the force of the surfce cting on the object. riction 1. riction must deend on the surfces in contct. Some surfces re more slier thn others. Surfce roerties re contined in quntit clled the coefficient of friction,.. riction must lso deend on the forces beteen surfces. Hev objects hve more friction thn light objects. The force roert is the norml force, N. This is onl n roimtion for friction. riction is ctull ver comlicted. This model is useful becuse it is firl es to use in our eercises. Kinetic riction If one surfce slides over nother, e re deling ith kinetic friction. The mgnitude of the kinetic friction fk is f k N k Since both the frictionl nd norml forces re mesured in the sme units, k is dimensionless. 4-6
Sttic riction If the surfces re t rest reltive to ech other, the sitution is more subtle. Assume force ushes on block tht remins t rest. Aling Neton s second l f f s s 0 m This is the obvious result tht if n object remins t rest under the influence of n eternl lied force, the sttic frictionl must ectl equl the lied force. So, if no eternl force is lied, there is no sttic friction force. If the eternl force is 1 N, so is the frictionl force. If it is N, so is friction. And so on. Eventull, the lied force becomes lrge enough to cuse the object to slide. The lrgest ossible frictionl force defines the coefficient of sttic friction f sn s, m In generl, the sttic friction is less thn the mimum vlue, fs f s,m Imortnt: e onl use the eqution fs sn in situtions here the object is bout to move. Look for ords like lrgest, smllest, bout to move, et l. If ou see those ords in the roblem, ou cn use the equlit for sttic friction. If ou do not see those ords, the sttic friction force hs to be determined from Neton s second l s I shoed ou just bove. Direction of riction (ges 104-105) The sttic frictionl force cts in htever direction necessr to revent the objects from beginning to slide or sli. Kinetic friction cts in direction tht tends to mke the sliding sto. If book slides to the left long tble, the tble eerts kinetic frictionl force on the book to the right, in the direction oosite to the motion of the book. rom Neton s third l, frictionl forces come in interction irs. If the tble eerts frictionl force on the sliding book to the right, the book eerts frictionl force on the tble to the left ith the sme mgnitude. 4-7
Tension An idel cord ulls in the direction of the cord ith forces of equl mgnitude on the objects ttched to its ends s long s no eternl force is eerted on it nhere beteen the ends. An idel cord hs zero mss nd zero eight. Problem-Solving Strteg for Neton s Second L (ge 113) Decide ht object ill hve Neton s second l lied to it. Identif ll the eternl forces cting on the object. Dr n BD to sho ll the forces cting on the object. Choose coordinte sstem. If the direction of the net force is knon, choose es so tht the net force (nd the ccelertion) re long one of the es, ind the net force b dding the forces s vectors. Use Neton s second l to relte the net force to the ccelertion. Relte the ccelertion to the chnge in the velocit vector during time intervl of interest. Problem 4.9 A 10.0-kg termelon nd 7.00-kg umkin re ttched to ech other vi cord tht rs over ulle. riction is negligible everhere in this sstem. () ind the ccelertion of the umkin nd the termelon. Secif mgnitude nd direction. (b) If the sstem is relesed from rest, ho fr long the incline ill the umkin trvel in 0.30 s? (c) ht is the seed of the termelon fter 0.0 s? Decide ht object ill hve Neton s second l lied to it. There re to objects, the termelon nd the umkin. 4-8
Identif ll the eternl forces cting on the object. On the umkin, the forces re: its eight, the norml force due to the left incline, nd the tension in the cord. On the termelon, the forces re: the eight of the termelon, the norml force from the right incline, nd the tension in the cord. The most generl roblem ould include frictionl forces, but e re told tht friction is negligible. Dr BD to sho ll the forces cting on the object. To objects, to free-bod digrms. N P T T N P Choose coordinte sstem. If the direction of the net force is knon, choose es so tht the net force (nd the ccelertion) re long one of the es, To different coordinte sstems re needed since there re to objects. In both cses, set the -is long the inclines. ind the net force b dding the forces s vectors. rom the BD, find the net force in the nd directions for ech object. Here is ho ou del ith inclined lnes. Dr the figure so tht the incline hs shllo ngle. Notice tht the eight mkes n ngle ith the negtive -is equl to the ngle of the incline. This mens sin cos 4-9
nd the -comonent is ssocited ith the sine function nd the -comonent is ssocited ith the cosine function. h? The ngle is mesured from the verticl. The forces for the umkin: T nd P N P P nd for the termelon: T nd N Use Neton s second l to relte the net force to the ccelertion. or the umkin: T P T m g sin 53 m m m N P m N P P g cos 53 m m 0 or the termelon: m g sin 30 T T m m m N m N g cos 30 m m 0 Relte the ccelertion to the chnge in the velocit vector during time intervl of interest. or some strnge reson, the termelon is tied to the umkin. This mens tht Since there is onl one ccelertion, let s cll the ccelertion of both fruit. 4-10
The motion in the -direction is given b T m g sin 53 m nd m g sin30 T m rom the first eqution, T m m g sin53 Substituting into the second eqution m g sin 30 ( m m g sin 53 ) m m g sin 30 m g sin 30 T m m g sin 53 ( m m m ) sin 30 m (10 kg)sin 30 (7 kg)sin 53 7 kg 10 kg 0.340 m/s m m sin 53 g (9.8 m/s ) ht does the negtive sign men? (b) Distnce trveled in 0.30 s is v 0 i 1 t 0.015 m 1 ( t) ( 0.340 m/s )(0.30 s) ht if the gourds re not frictionless? ht is the coefficient of friction needed to kee them from moving? (c) The seed of the termelon (nd the umkin since the re connected together) v f v v i f t v i t 0 0.340 m/s 0.0s 0.068m/s All roblems follo this formt. 4-11
Problem-Solving Strteg for Neton s Second L (ge 113) Decide ht object ill hve Neton s second l lied to it. Identif ll the eternl forces cting on the object. Dr n BD to sho ll the forces cting on the object. Choose coordinte sstem. If the direction of the net force is knon, choose es so tht the net force (nd the ccelertion) re long one of the es, ind the net force b dding the forces s vectors. Use Neton s second l to relte the net force to the ccelertion. Relte the ccelertion to the chnge in the velocit vector during time intervl of interest. or roblems here ou kno bout the motion. or emle, n object slides long surfce. Define sstem Crete BD Use Solve equtions rom BD rom motion or roblems here ou nt to determine something bout the motion. or emle, ho fr ill the object trvel if it strts from rest? Define sstem Crete BD Use rom BD To constnt ccel equtions Solve equtions All the roblems cn be solved b creful use of this sstem. It tkes rctice to mster. 4-1
Arent eight hen ou ride in roller coster, sometimes ou feel hevier thn norml, sometimes lighter. or n elevtor ccelerting u, ou feel hevier, ig. 04.55 ht hens hen the elevtor ccelertes donrd? Air Resistnce Too hrd for us to ork ith quntittivel. undmentl orces Grvit onl ttrctive. Long rnge force. Ver ek. Electromgnetism elined in volume Strong holds nucleus together ek seen in some rdioctive dec rocesses 4-13