Newton s Three Laws. d dt F = If the mass is constant, this relationship becomes the familiar form of Newton s Second Law: dv dt

Transcription

1 Newton s Three Lws For couple centuries before Einstein, Newton s Lws were the bsic principles of Physics. These lws re still vlid nd they re the bsis for much engineering nlysis tody. Forml sttements of Newton s Three Lws re given below. Informl eplntions of Newton s Three Lws re given below ech forml sttement. Newton s First Lw: An object t rest tends to sty t rest nd n object in motion tends to sty in motion with the sme speed nd in the sme direction unless cted upon by n unblnced eternl force. Inerti is property of mtter tht resists chnges in motion. If mss is not moving, it will sty tht wy until n unblnced eternl force strts to move it; if mss is in motion, it will sty in motion with the sme speed nd direction until n unblnced eternl force chnges its motion chrcteristics (friction could slow it down, or force could ccelerte its motion). For emple, let us consider hockey puck on the ice (ssume the ice is perfectly level nd frictionless). If the puck is plced down on the ice, it will sty motionless until someone hits it with stick or skte becuse of its inerti. Also due to inerti, when slpped, the puck will tend to move in stright line with constnt speed until n eternl force (such s nother plyer, or the golie, or the net) chnges its motion. As second emple of Newton s First Lw, consider cr ccelerting from stoplight. As the cr ccelertes from zero motion, your body tends to push bck into the set due to its inerti (trying to remin t rest). Also, s the cr is brked from high speed bck to stopping, your body is flung forwrd due to its inerti in motion. Hopefully you hve your setbelt on, or else Newton s First Lw could hve bd consequences. Newton s Second Lw: The ccelertion of n object s produced by net force F is directly proportionl to the mgnitude of the net force, in the sme direction s the net force, nd inversely proportionl to the mss m of the object: F = m. A resultnt eternl force F cting on body will ccelerte tht body in the direction of F, with ccelertion = F/m. Accelertion is the second time rte of chnge of position, lso the first time rte of chnge of velocity; ccelertion is to velocity wht velocity is to position. Newton s originl sttement of the Second Lw ws tht the resultnt eternl force F is equl to the time rte of chnge of momentum (mv, mss times velocity): F = d dt ( mv) If the mss is constnt, this reltionship becomes the fmilir form of Newton s Second Lw: dv F = m = m dt

2 Before Newton developed his Second Lw, the previling belief ws tht force ws proportionl to velocity: F = mv. This ppered to be true for the motion of horse-drwn crts, since friction domintes this problem. Newton revolutionized engineering mechnics; his lws were unchllenged until Einstein s Reltivity work. Newton s Lws re still the bsis for most engineering dynmics tody. Newton s Third Lw: For every ction, there is n equl nd opposite rection. This lw is fmilir in everydy situtions; force cnnot be pplied to n object unless something resists the rection of tht force. In order to wlk cross the floor, you must push bck on the floor with your foot; then, ccording to Newton s Third Lw, the floor pushes forwrd on your foot, which propels you forwrd. This, of course, requires friction to work. If free-floting stronut were to throw bsebll, there is nothing to resist the throwing force, so s the bsebll ccelertes in the direction of throwing, the stronut would ccelerte bckwrds, with force equl nd opposite to the throwing force. The stronut would ccelerte t much smller level (by Newton s Second Lw) since her mss is much greter thn the bsebll s mss. The recoil of gun during firing is nother emple of Newton s Third Lw. As finl emple, if person ttempts to jump to dock from smll silbot, they my end up lnding in wter if they do not understnd Newton s Third Lw: similr to the stronut emple, the jumping force of the humn on the bot will tend to push the bot bckwrds; the equl nd opposite force of the bot on the humn will propel tht person towrd the dock, but since the bot moves bckwrds, the person my end up wet. The sme problem eists for lrge silbots, ecept with lrger bot inerti, it is less noticeble.

3 Newton s st Lw Emple: Block on frictionless surfce To demonstrte in softwre Newton s First Lw (the lw of inerti) we consider point-mss block free to slide on frictionless, perfectly flt XY plne. Grvity is norml to this plne, in the Z direction, nd hence grvity does not enter into this problem due to our frictionless surfce ssumption (ecept for the fct tht grvity keeps the block in the XY plne). When the block is plced t rest on the plne with no net unblnced eternl force, it will remin t rest due to the block s property of inerti. The user cn then strt motion by pushing on the block with the hptic interfce try single X or Y directions first, but both cn occur t once. The user feels the inerti, which is the block s resistnce to chnge in motion. Once the block is in motion, if the force is removed, it will tend to remin in motion t the velocity (speed nd direction) obtined when the force stopped. A reverse force is required to return the block to rest. The user feels the block s inerti in motion, gin resisting chnge in motion. With the frictionless ssumption, it is not esy to return the moving block to rest with the joystick. For the simulted motion when force is pplied, we use Newton s nd Lw F = m to solve for the resulting ccelertion. The free-body digrm (FBD) is simple in this cse, shown for one direction X below: W F m The resulting ccelertion is simply = F /m, while the force is pplied (for Y motion, y = F y /m). We cn use kinemtics to determine the resulting motion; we strt with the known, constnt ccelertion nd integrte twice (the constnts of integrtion re zero if we strt from rest nd mesure the displcement from the initil position). The left epressions re for the motion when the force is pplied; the right epressions re for the constnt velocity motion phse when the force hs been removed. The below epressions re for X motion, but pply eqully (independently but simultneously) to Y motion. v F () t = m () t = 0 () t = t v() t = vc t () t = + v t () t = 0 C In the right epressions bove, v C is the constnt velocity chieved t the instnt the force is removed: v C () t = tmax, where t MAX is the time when the force is removed. Also, 0 is the displcement chieved in the first motion phse, t MAX 0 =. N

4 Inertil force is defined to be F0 = m in the X direction (similr for Y). Here force is in quotes since it hs units of force, N, but it is not true force, rther n effect of ccelertion tht cn be felt. Inertil force gives the user the feel of the mss s inerti during the simultion. User inputs: Forces F nd F y vi the hptic interfce (directions commnded by user, mgnitudes ll ssumed to be 0 N). Computer sets: Visulize: m = 0 kg point mss in motion, plus kinemtics plots for, v, ; lso y, v y, y Numericl Disply: Nothing. User Feels: Block inertil forces F0 = m in X nd F0 y = my in Y; feels nothing during constnt velocity motions. Emple: When the user pushes with F = 0 N nd F y = 0 for three seconds (t MAX = 3 sec), followed by F = F y = 0 for three more seconds, the numericl results re: First motion phse: = m/s Second motion phse: v C = 3 m/s, 0 = 4.5 m The ssocited kinemtics plots vs. time re shown below:

5 v time Kinemtics Plots for motion in the X direction, Newton s First Lw Simultion In the first three seconds when the constnt force is pplied to the block, the resulting constnt ccelertion (from Newton s Second Lw) is = m/s ; in this motion rnge, the block velocity increses linerly nd the block s position increses prboliclly. In the remining three seconds of motion, the constnt force hs been removed so there is constnt velocity (v C = 3 m/s, the mimum vlue from the first three seconds) ccording to Newton s Second Lw. In the constnt velocity motion rnge, ccelertion is zero, nd position linerly increses from its previous ending vlue of 0 = 4.5 m. You cn see tht the position chnges by 3 m every sec, which is the constnt velocity of 3 m/s.

6 Newton s nd Lw Emple: Two-mss ccelertion In this softwre simultion, two msses re connected by n idel cble (mssless, perfectly stiff) s shown in the digrm. For this point-mss problem, we ignore the pulley rottionl inerti. The system is relesed from rest nd we wish to clculte the resulting system motion using Newton s nd Lw. The dynmic coefficient of friction between m nd the motion surfce is µ. The free-body digrms, one for ech point mss, re shown following the system digrm. Y m X y m W T F f m T m N W For ech free-body digrm, we pply Newton s nd Lw, F = ma i i. For mss, this vector eqution yields one eqution in the X nd one in the Y direction; for mss, only one eqution, in the Y direction, results: X : T F f Y : N W = m = m y Y : W T = m y

7 The sme cble tension T is pplied to ech mss, by Newton s 3 rd Lw (since we re not considering pulley dynmics). The friction force is F f = µ N, opposing the direction of motion; N is the norml force of the motion surfce cting on mss. The weight forces re W ccelertion vectors re: i = m g, i=,. The i A = y = 0 A = y = 0 Tht is, mss does not ccelerte in the Y direction nd mss does not ccelerte in the X direction; lso, since the point msses re connected by n idel cble, their ccelertions re the sme: = y =. Using this informtion, the bove three dynmics equtions of motion become: T µn = m N m g = 0 m g T = m Substituting the Y eqution ( N = m g ) into the X eqution for mss yields: T = ( + µ g) m. Further substituting this cble tension for T in the Y eqution for mss, we cn solve for the system ccelertion : ( m µ m ) g = m + m Hving solved for the ccelertion using Newton s nd Lw, we cn use kinemtics to determine the resulting motion; we strt with the known, constnt ccelertion nd integrte twice (the right epressions ssume zero initil conditions since the system ws relesed from rest nd we cn mesure the displcements from the initil loctions): v () t = () t = () t = v 0 + t v() t = t () t = t 0 + v y0t + () t t = The Y ccelertion, velocity, nd displcement epressions for mss re identicl, where the positive indictes downwrd motion here, strting from the initil displcement of zero. Note: If µ m m, the bove eqution for predicts tht the ccelertion is zero (for =) or negtive (for >). In this cse there will be no motion since the combintion of µ nd m re too lrge for the given m. User sets: msses m nd m, plus dynmic coefficient of friction µ: 0 m 0 0 m 0 0 µ

8 Computer sets: Visulize: g = 9.8 m/s, in the Y direction. point msses in motion, plus kinemtics plots for, v, (sme s y). Numericl Disply:, T, F f User Feels: Inertil forces m or m (user chooses), opposite to direction. Emple: When the user enters m = 0 kg, m = 5 kg nd µ = 0. 5, the numericl results re: =.9 m/s ; T = 37.6 N; F f = -4.7 N The ssocited kinemtics plots vs. time re (this simultion ws run for n X (nd Y) displcement of 0 f m, with finl time of t 0 f = = =. 96 sec): v time Kinemtics Plots for m in the X direction (identicl to m in the Y direction) The resulting constnt ccelertion (from Newton s Second Lw) is =.9 m/s ; the block velocity increses linerly from zero nd the block s position increses prboliclly. The sme ect plots pply to Y motion, where the displcement is down, mesured from the initil loction of the second mss.

9 Comprehension Assignment: Once you get the feel for this simultion, run the progrm severl times to collect nd plot dt: for fied vlue of the dynmic coefficient of friction µ, vry the mss rtio m /m over its llowble rnge nd determine the resulting ccelertion. Plot vs. m /m. Repet these plots for vrious vlues of µ over its llowble rnge. Discuss the trends you see do the results mke sense physiclly?

10 Newton s 3 rd Lw Emples There re two simultions in the softwre to demonstrte Newton s Third Lw. The first one is simply liner spring tht my be compressed (or etended) vi the hptic interfce. The user then feels the equl nd opposite force of the virtul spring pushing (or pulling) on the hnd. A liner spring obeys Hooke s lw: F = K, where K is the spring constnt (N/m) nd is the mount the spring is stretched from its neutrl position. As the user moves the joystick to compress the virtul spring, the spring feels the force nd moves ccordingly. At the sme time, force pushes on the user s hnd, of the sme mgnitude, but reversed in direction. Thus, the force of the user cting on the spring is equl nd opposite to the force of the spring cting on the user. This equl nd opposite force eists whether the user compresses or etends the spring. Try it nd you will see nd feel the effects of Newton s Third Lw. The second simultion involves projectile fired from cnnon on the wter in ttempt to hit trget on lnd. There re two cnnons; one is rigid bse, fied to dock whose pilings re sunk deep into bedrock under the wter. The second is cnnon mounted to free-floting bot. Both cnnons hve the sme shooting ngle nd the sme initil muzzle velocity, clculted to successfully hit the trget from fied-bsed cnnon. Try the first, fied-bsed, cnnon nd you will see the trget is hit. By Newton s Third Lw, there will be n equl nd opposite rection force from the cnnon bck on its bse. When the bse is fied, this rection force will be resisted successfully, in this cse by the fied pilings. When you try the second, bot-mounted cnnon you will see the cnnon projectile flls short of the trget. In this cse, the equl nd opposite rection force from firing the cnnon cuses the bot to move wy from the shore. Unless new cnnon ngle nd muzzle velocity is clculted for this cse, the projectile will fll short of the trget due to Newton s Third Lw.