Physics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle
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1 Chaper 2 Newonian Mechanics Single Paricle In his Chaper we will review wha Newon s laws of mechanics ell us abou he moion of a single paricle. Newon s laws are only valid in suiable reference frames, and we will discuss wha makes a reference frame a suiable reference frame. We will also review he various conservaion laws you should have already encounered in your inroducory physics course. In addiion, we will discuss he limis of Newonian mechanics. Newon s Laws The following hree laws of Newonian mechanics should have been discussed in your inroducory physics course: Newon s Firs Law: A body remains a res or in uniform moion unless aced upon by a force. This law does no ell us very much abou he concep of force, excep wha we mean wih zero force: if we see a body a res or in uniform moion, we know ha he force acing on he body is zero. Newon s Second Law: A body aced upon by a force moves in such a manner ha he ime rae of change of is linear momenum equals he force. Newon defined he linear momenum of a paricle of mass m moving wih a velociy v as mv. The second law can hus be used o define he force F = d(mv)/d. This definiion is of course only useful if he mass and he velociy of a paricle are defined. Newon s Third Law: If wo bodies exer forces on each oher, hese forces are equal in magniude and opposie in direcion. This law is only rue if he force acing beween he bodies is direced along he line connecing he bodies (hese forces are called cenral forces). Forces ha are velociy dependen are in general non-cenral forces and do no saisfy Newon s Third Law. Conservaion of linear momenum is a direc consequence of he hird law. If F 1 = -F 2 hen d(m 1 v 1 )/d = - d(m 2 v 2 )/d. This equaion can be rewrien as d(m 1 v 1 + m 2 v 2 )/d = or m 1 v 1 + m 2 v 2 = consan. Reference Sysems Newon s laws are only valid in an appropriae reference frame. We can use his requiremen o define inerial reference sysems: an inerial reference frame is a reference frame in which Newon s laws are valid. If Newon s laws are valid in one reference frame, hey are also valid in any reference frame in uniform moion wih respec o he firs frame. In order o be able o describe a free paricle (a paricle on which no force is acing) in a reference frame, he reference frame mus saisfy he following condiions: - 1 -
2 The equaion of moion of he paricle should be independen of he posiion of he origin of he coordinae sysem. The equaion of moion of he paricle should be independen of he orienaion of he coordinae sysem. Time mus be homogeneous (he velociy of a free paricle mus be consan). Single-Paricle Moion If we know he force acing on he paricle, we can use Newon s second law o describe is moion: dv d = 1 m F Noe: F mus be he oal force acing on he paricle. As long as he force is consan, we can usually obain an analyical expression for he velociy and/or he rajecory of he paricle. The siuaion becomes more complicaed when he force is ime dependen, velociy dependen, and/or posiion dependen. In hose circumsances we may need o rely on numerical mehods o predic he moion of he paricle. The second law can be rewrien as ( ) v ( ) = d dv = v + d The velociy a ime + d can hus be deermined from he velociy a ime using he following relaion: ( ) = v ( ) + d v + d This relaion can be used o deermine he velociy as funcion of ime, if we know 1) he velociy a one specific ime and 2) he force F is known (bu does no need o be consan). Once we know he velociy as funcion of ime, we can deermine he posiion as funcion of ime: m F m F r ( + d) = r () + v ( )d Le us illusrae he use of numerical mehods by focusing on projecile moion. If he graviaional force is he only force presen, we can express he moion of he paricle analyically. By comparing he resuls of a numerical calculaion wih he analyical soluion, we can sudy he limiaions of he numerical approach. Since he graviaional force is acing in he - 2 -
3 verical direcion (along he y axis), he force will have no effec on he moion in he horizonal direcion (along he x axis): v y v x ( + d) = v x ( ) ( + d) = v y ( ) gd x( + d) = x( ) + v x ( )d ( ) = y( ) + v y y + d Many differen programs can be used o sudy he evoluion of hese equaions. No maer wha approach is being used, he mos criical choice he user will have o make is he size of he sep size d. In he case of a consan force, he expressions for he velociy a ime + d are correc, independen of he sep size d. However, he expressions for he posiion a ime + d are only correc if he velociy is consan over he period beween ime and ime + d. This is a reasonable approximaions if he sep size d is small, bu for a large sep size d his is clearly a poor approximaion (and large errors will resul). A very small sep size will increase he compuing ime and may lead o rounding errors. On he Physics 235 homepage you can find an example of a sudy of projecile moion using Excel. Using he seup sored in he file ProjecileMoion.xls we can sudy he effec of he choice of he sep size d. Consider he case of projecile moion, saring a ime = s a he origin of our coordinae sysem wih a velociy of +7 m/s in he horizonal and +7 m/s in he verical direcion. Figure 1 shows a comparison beween he resuls of he analyical calculaion of he rajecory of he projecile (blue daa poins) and he resuls of he numerical calculaion of he rajecory (red daa poins) wih a ime sep of 1 s. The difference beween hese wo are shown by he green daa poins (defined as y numerical y analyical ). There is clearly a significan difference beween he numerical and he analyical calculaion. Figure 2 shows he resuls of he same calculaion as shown in Figure 1, excep ha he ime sep was changed o 1 s. As a resul, here are clearly more daa poins on he rajecory. However, he more imporan difference is he significan reducion of he difference beween he analyical and he numerical calculaions. Figure 3 shows anoher resul of he projecile moion, now obained wih a sep size of.25 s. The difference beween he numerical and he analyical mehod is furher reduced. The differences beween he numerical mehod and he analyical mehod a a horizonal disance of 1, m are shown in he following able for he sep sizes used o generae Figures 1-3. ( )d d (s) Difference (m)
4 Projecile Moion Theory 3.E+4 2.5E+4 Numerical Verical Difference Verical Disance (m) 2.E+4 1.5E+4 1.E+4 5.E+3.E+.E+ 5.E+4 1.E+5 1.5E+5 Horizonal Disance (m) Figure 1. Resuls of numerical and analyical calculaions of projecile moion wih d = 1 s. Verical Disance (m) 3.E+4 2.5E+4 2.E+4 1.5E+4 1.E+4 5.E+3.E+ Projecile Moion.E+ 5.E+4 1.E+5 1.5E+5 Horizonal Disance (m) Theory Numerical Verical Difference Figure 2. Resuls of numerical and analyical calculaions of projecile moion wih d = 1 s. Verical Disance (m) 3.E+4 2.5E+4 2.E+4 1.5E+4 1.E+4 5.E+3 Projecile Moion Theory Numerical Verical Difference.E+.E+ 5.E+4 1.E+5 1.5E+5 Horizonal Disance (m) Figure 3. Resuls of numerical and analyical calculaions of projecile moion wih d =.25 s
5 Now le us consider wha happens when besides he graviaional force, here is a drag force acing on he paricle. The drag force is usually proporional o he power of he velociy and direced in a direcion opposie o he direcion of moion. The ne force on he paricle is his equal o F x = kmv x F y = kmv y mg Since he force is known, we can deermine he velociy of he paricle: v y v x ( + d) = v x ( + d) = v y ( ) + d ( ) + d m F x = v x ( ) + d ( m kmv x ( ) ) = ( 1 kd)v x m F = v y y ( ) + d ( m kmv y ( ) mg) = ( 1 kd)v y ( ) ( ) gd The posiion of he paricle can be found once we have deermined he velociy as funcion of ime: ( ) = x( ) + v x x + d ( ) = y( ) + v y y + d In principle, we can sill solve his problem analyically (see Example 2.5 in he ex book) bu we will use he same numerical approach as we used in our sudy of projecile moion wihou drag o sudy he rajecory for differen values of k. The resuls of a calculaion for k =.1 and wih ime seps of.25 s is shown in Figure 4. Verical Disance (m) 1.4E+4 1.2E+4 1.E+4 8.E+3 6.E+3 4.E+3 Projecile Moion wih Drag ( )d ( )d Wihou Drag Wih Drag 2.E+3.E+.E+ 5.E+ 3 1.E E+ 4 2.E E+ 4 Horizonal Disance (m) 3.E E+ 4 Figure 4. Projecile moion wih drag
6 Conservaion Laws Several imporan conservaion laws are a direc consequence of Newon s laws of moion. These conservaion laws can significanly reduce he effor required o solve cerain mechanics problems. In his Secion we will briefly discuss he mos imporan conservaion laws ha we will use in classical mechanics. The oal linear momenum p of a paricle is conserved when he oal force on i is zero. This law is a direc consequence of Newon s second law, which relaes he change in he linear momenum of a paricle o he force acing on i. The angular momenum L of a paricle subjec o no orque is conserved. This law is a direc consequence of he definiion of angular momenum and orque. In fac one can argue ha orque was defined such ha i is equal o he rae of change of he angular momenum (dl/d). The oal energy E of a paricle in a conservaive force field is consan in ime. The oal energy E is defined as he sum of he kineic energy T and he poenial energy U. The poenial energy U is defined by he force field only o wihin a consan. I has no absolue meaning, and only differences in he poenial energy are physically meaningful. The force field is conservaive if he line inegral of he force beween wo poins is pah independen. In his case, we can wrie he force as he gradien of a scalar funcion, and his scalar funcion is he poenial energy U: F = U We can show ha he rae of energy change de/d will be zero if he poenial energy U does no depend explicily on ime ( U/ = ). These hree conservaion laws are he mos imporan conservaion laws in classical mechanics and we will use hem in many differen applicaions. An imporan applicaion of one of our conservaion laws is he predicion of moion based on a poenial energy curve U(x). Consider for example, he poenial energy curve shown in Figure 5. Since he oal energy E is he sum of he poenial energy U and he kineic energy T, he oal energy will always be larger or equal o he poenial energy U. Looking a Figure 5, we can immediaely draw some imporan conclusions: No paricle can exis wih a oal energy E less han E. A paricle wih energy E 1 can only be presen beween x a and x b. A paricle wih energy E 4 can be presen a any posiion
7 Since he force is relaed o he derivaive of he poenial energy, he posiions where he derivaive is equal o are he posiions where he ne for he on he paricle is zero (hese are he equilibrium posiions). Figure 5. Poenial energy U(x) as funcion of posiion. Consider he poenial energy in he viciniy of an equilibrium posiion, and assume we have chosen our coordinae sysem such ha he equilibrium posiion corresponds o x =. We can expand he poenial around he equilibrium poin: ( ) = U + x du U x dx d 2 U 2! dx 2 + x2 d 3 U 3! dx 3 Since a he equilibrium poin, he slope of U(x) is zero (du/dx = ) and since we can define he poenial o be zero a his equilibrium poin, we can rewrie he expansion of U as U x ( ) = x2 d 2 U 2! dx 2 + x3 + x3 d 3 U 3! dx 3 For small displacemens wih respec o he equilibrium posiion, x is small. As a resul, he firs non-zero erm in he expansion will dominae he expansion: U x ( ) = x2 d 2 U 2! dx For he equilibrium o be sable, he poenial energy on eiher side of he equilibrium poin mus be higher han he poenial energy a he equilibrium poin (which we defined o be ). In order o achieve his we mus require ha - 7 -
8 d 2 U dx 2 > If his condiion is no saisfied, he equilibrium is an unsable equilibrium. Limiaions of Newonian Mechanics Newonian mechanics can be used o describe many every-day macroscopic phenomena. When we sudy macroscopic phenomena, we can measure boh he posiion and he linear momenum of objecs of ineres wih grea precision. However, when we sar o sudy microscopic objec we discover ha we can no longer measure he posiion and he linear momenum wih grea precision. In fac, he acual measuremen may influence he sae of he sysem. In his regime, our capabiliy of deermining he posiion and he linear momenum of an objec are limied by he Heisenberg uncerainy principle, which saes ha ΔxΔp 1 34 If we measure he posiion wih infinie precision, he uncerainy in he linear momenum approaches infiniy. In his regime, Newonian mechanics can no longer be used, and we need quanum mechanics o describe microscopic sysems. The limiaions of Newonian mechanics also appear when we sudy moion wih velociies close o he speed of ligh. In his regime, we need he heory of relaiviy. One fundamenal assumpion in he heory of relaiviy is ha he speed of ligh is consan, he same in each reference frame. This is clearly inconsisen wih Newonian mechanics, and he rules ha govern ransformaions of posiion and velociy beween coordinae sysems. Anoher limiaion of Newonian mechanics becomes obvious when we ry o describe sysems wih large numbers of paricles. Even if we know all of he deails of he ineracion beween he paricles, is becomes very difficul o predic he properies of he sysem by carrying ou calculaions involving he each individual ineracion beween all he paricles. Such sysems can be described by heory of saisical mechanics, which relaes he properies of microscopic ineracions o he average macroscopic properies of he sysem. Js - 8 -
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