Study on Application of New Theorem of Kinetic Energy

Transcription

1 Intenational Jounal of pplied Physics and Mathematics Study on pplication of New Theoem of Kinetic Enegy Yong Yang * Tongchuan Banch of Shanxi Radio and TV Univesity, Tongchuan Polytechnics, Tongchuan New Distict, Shanxi, 7703, PR China. * Coesponding autho. Tel.: ; y.y.886@63.com Manuscipt submitted May 0, 05; accepted Mach 5, 06. doi: /ijapm bstact: New theoem of kinetic enegy (NTKE) gets id of the impefection that lies in the classical theoem of kinetic enegy (CTKE), i.e. CTKE fails to eveal coectly that both the non-consevative foce and non-consevative eacting foce do dissipated wok simultaneously in the equal amount, and both convet the mechanical enegy of a system into non-mechanic enegy of the foce-beaing object. Stating with undestanding the meaning of NTKE, the autho makes attempts to apply NTKE on paticles and on paticle system to wok out pocedues solving physical poblems, and methods to calculate tems in the new theoem and eveal the physical impotance of calculated esults. Key wods: New theoem of kinetic enegy on paticles, new theoem of kinetic enegy on paticle system, solving pocedues, calculation of implicated wok, calculation of dissipated wok.. Intoduction Mechanical wok is a measue of enegy tansfeing and convesion. The basic concept in the new wok-enegy theoy of paticle dynamics includes: both consevative and non-consevative foces do non-dissipated wok in such a way that the mechanical enegy is tansfeed between paticles inteacting; in addition, non-consevative foce does dissipated wok to convet mechanical enegy into non-mechanical enegy of foce-beaing paticles; in geneal, non-consevative foce does dissipated and non-dissipated wok; non-consevative foce does dissipated wok and non-consevative eacting foce also does dissipated wok in the equal amount. The ule that foce does mechanical wok is expessed as new wok-enegy theoy [] which can objectively eveal the pocess of mechanical wok and enegy tansfeing and convesion as well as thei elation, anothe upgading of classical wok-enegy equation. New theoem of kinetic enegy (NTKE), as a fundamental equation fo new wok-enegy theoy, must be studied and undestood coectly befoe it is coectly applied to explain objective phenomena, povide guidance to pactices and pedict futue.. Study on pplication.. New Theoem of Kinetic Enegy on Paticles The NTKE on paticles is: in the motion pocess of Paticle m i, its incement of kinetic enegy (infinitesimal) equals to the sum of total wok implicated wok c (infinitesimal) done by consevative foce, total (infinitesimal) done by non-consevative foce, total dissipated wok (infinitesimal) nci done by non-consevative foce, and total dissipated wok (infinitesimal) done by non-consevative ncd 3 Volume 6, Numbe, pil 06

2 Intenational Jounal of pplied Physics and Mathematics eacting foce. It is expessed in the equation below: The diffeential fomula: d c +d nci +d ncd +d ncd = de k () The integation fomula: c + nci + ncd + ncd = Ek - E k0 () In applying NTKE to solve pactical poblems, we need not only to study the foces applied to the subject but also to undestand motion of each foce-applying paticle; in addition, we shall be able to apply coectly the new definition of mechanical wok in New Wok-Enegy Theoy to compute evey tem in the equation of the NTKE. Impotance of NTKE is evealed in the study on applications below... Study on pplication of NTKE on Paticles NTKE on paticles enables us to chaacteize tansfeing and convesion of mechanical enegy in both moving and static paticles. Fig.. nalyze foces applied to the wooden block. Example : as shown in Fig., a hoizontal flat plate in mass of M is fixed on the gound; ove the plate, a wooden block in mass of m is moving towads ight in staight line at a unifom speed and the coefficient of sliding fiction of the wooden block against the hoizontal flat plate is μ. Question: calculate the wok done by non-consevative foce on Paticles and when the wooden block moves to ight fo the distance s. nswe: using the gound as efeence, establish a coodinate system 0-xy shown in Fig.. Since the wooden block moves hoizontally and the plate is still, both can be consideed as paticles. Because thee is dissipating foce doing wok, it is solved using the NTKE on paticles. The objects inteacting against each othe ae numbeed as below: : the wooden block; : the flat plate; 3: the eath (gound); 4: the object applying pull The meaning of the ight subscipt of the foce F ij applied to paticles: the fist place i is the numbe of the foce-baing objects and the second place j is the numbe of the foce-applying objects (i, j =,, 3, 4; and i j ). ) Fist, use the motion paticle as the subject to exploe the elations about wok-enegy tansfeing and convesion. nalyze foces applied to Paticle : The gavity G 3, the suppoting foce N by the flat plate, the fiction f against the flat foce and the pull f 4, the last thee of which ae non-consevative foce. Thei espective eacting foces ae, in tun: the gavity G 3 fom the paticle, the pessue N and the fiction f against Paticle, and the pull f 4 against the foce-applying object. 3 Volume 6, Numbe, pil 06

3 Intenational Jounal of pplied Physics and Mathematics Since Paticle moves in a staight line at a unifom speed, the equation of Newton's Second Law fo Paticle is: G + N + f + f = ma a =0 3 4 (3) The equations set consisting of espective component equations and auxiliay equations is: 4 Fx j f F4 0 j f f μn N N G3 mg 4 Fyj N G3 0 j (4) Solve the above-stated equation set to obtain: N N G3 mg f4 f f μmg (5) In the pocess that Paticle moves along the positive diection of xis x in the distance s, the wok done by foces on Paticle is: The wok done by non-consevative foces is to be calculated with the classical definition of the mechanical wok: integate the poduct of the consevative-foce vecto point and the infinitesimal of the absolute displacement made by the foce-beaing paticle [], which is a non-dissipated wok. 0 si x0 s x0 s c c3 G d G3 cos d 0d x 0 x (6) 0 Then calculate the woks of Paticle done by non-consevative foces, which shall be calculated in accodance with the new definition of wok and include the dissipated wok and the implicated wok. Calculate the implicated wok by the non-consevative foce: integate the poduct of the vecto point of the non-consevative foces and the infinitesimal of the implicated displacement of the foce-beaing paticle [3] (the absolute displacement of the foce-applying paticle), which is a non-dissipated wok. 0 = N d (7) nci() 0 0 = f d (8) nci() 0 40 si x40 s nci4 cos0 ( ) x (9) 40 f d f dx f s μmgs 33 Volume 6, Numbe, pil 06

4 Intenational Jounal of pplied Physics and Mathematics Substitute Equations (7)-(9) into the equation below, to obtain the total implicated wok of Paticle done by non-consevative foces, as below: 4 nci ncij j nci() nci() nci3 nci f s 4 f s( μmgs) 4 (0) Calculate the dissipated wok by the non-consevative foce: integate the poduct of the vecto point of the non-consevative foces and the half of the infinitesimal of the elative displacement of the foce-beaing paticle elative to the foce-applying paticle [4], [5]. () N d ncd N dx dx 0 si x0 s x0 s cos x 0 x () 0 ncd() f d f dx μmg dx μmgs 0 s x0 s x0 s cos ( ) 0 x 0 x () ncd 4 f d 4 4 (3) 40 Thee is no elative displacement of Paticle elative to the pull-applying point. Substitute Equations ()-(3) into the equation below, to obtain the total dissipated wok of Paticle done by non-consevative foces, as below: 4 ncd ncdj j ncd() ncd() ncd3 ncd ( μmgs) μmgs (4) The above-stated equation epesents the dissipated wok of the paticle by non-consevative foces, and though the pocess the mechanical enegy is conveted into the themal enegy and the defomation enegy in the paticle to which foces ae applied. Calculate the dissipated wok by non-consevative eacting foces Fom the definition fomula of the dissipated wok below [5] dncdij fij dij ( f ji ) dji f ji dji dncdji (5) It is known that, the dissipated wok of the foce-applying paticle done by non-consevative eacting 34 Volume 6, Numbe, pil 06

5 Intenational Jounal of pplied Physics and Mathematics foces equals to the dissipated wok of Paticle done by coesponding non-consevative foces, each is: ncd () ncd () 0 (6) ncd () ncd () μmgs (7) The above-stated equation epesents the dissipated wok of the paticle by non-consevative eacting foces, and though the pocess the mechanical enegy is conveted into the themal enegy and the defomation enegy in the paticle to which foces ae applied. ncd 4 ncd 4 0 (8) Substitute Equations (6) ~ (8) into the equation below, to obtain the total dissipated wok of foce-applying paticles done by non-consevative eacting foces, as below: 4 ncd ncdj j ncd () ncd () ncd 3 ncd 4 ncd() ncd() ncd3 ncd4 4 j ncd ncd j μmgs (9) That is to say, the total dissipated wok total dissipated wok ncd of the paticle by non-consevative foces equals to the ncd of all foce-applying paticles by non-consevative eacting foces. Calculate the incement of kinetic enegy in Paticle Paticle makes linea motion at an unifom speed: Ek Ek0 mv0 Ek Ek Ek0 0 (0) Use equations to solve Substitute Equations (6), (0), (4), (9) and (0) into the Equation () - the NTKE fo paticles - to obtain: f4 s ( μmgs) ( μmgs) 0 () The ight pat to the equation's equal sign indicates: the incement of kinetic enegy in Paticle is zeo. The left pat to the equation's equal sign indicates: total mechanical enegy (value: μmgs ) of Paticle 35 Volume 6, Numbe, pil 06

6 Intenational Jounal of pplied Physics and Mathematics conveted fom the non-dissipated wok of Paticle by pull, half of which, due to the dissipated wok (the nd tem) done by non-consevative foces, is conveted into the non-mechanical enegy (value: μmgs ) of the foce-beaing paticle, and anothe half, due to the dissipated wok (the 3d tem) of the foce-applying paticle by the non-consevative foces, is conveted into the non-mechanical enegy [6] (value: μmgs ) of the foce-beaing paticle. Simultaneously when the mechanical enegy is pogessively tansfeed to the paticle, it is conveted into non-mechanical enegy of objects with mutual fiction instantly in equal amount; and the tansfeed mechanical enegy steam and the conveted non-mechanical enegy steam ae expessed with sepaate tems so as to show distinct coesponding elationship and clea implications of each tem. Since it obseves to the enegy convesion and consevation law, the ZTKE on paticle applies univesally in the paticle dynamics in the geneal condition that both the consevative foce and the non-consevative foce ae pesent. What if Paticle makes a linea motion at a vaying speed athe than at an unifom speed? When Paticle is acceleating: E E E () k k k0 0 Then f4 s ( μmgs) ( μmgs) 0 (3) It indicates that, of the total mechanical enegy of Paticle tansfeed fom the non-dissipated wok of Paticle (the st tem) by the pull, some is conveted into non-mechanical enegy (values ae μmgs, espectively) of Paticles and in mutual fiction, and the est is tansfeed to Paticle which is in motion esulting in inceased kinetic enegy in Paticle. When Paticle is deceleating: E E E k k k0 0 (4) Then f4 s ( μmgs) ( μmgs) 0 (5) It indicates that a pai of fiction foces makes dissipated wok to convet the mechanical enegy into non-mechanical enegy of Paticles and in mutual fiction, the value of both being μmgs, and some of the conveted mechanical enegy comes fom the non-dissipated wok (the st tem) of Paticle by the pull being tansfeed to Paticle and the est is at the cost of deceased kinetic enegy in Paticle in motion. ) Then, use the static paticle as subject to discuss the elationship in mechanical enegy 36 Volume 6, Numbe, pil 06

7 Intenational Jounal of pplied Physics and Mathematics tansfeing and convesion. s shown in Fig.. Fig.. nalyze foces applied to the flat plate. nalyze the foces applied to Paticle : the gavity G 3, the gound suppoting foce N 3, the binding foce T 3, the fiction foce f against Paticle, and the positive pessue N fom Paticle, of which the last fou ae non-consevative foce. Since Paticle is still, the equation of Newton's Second Law on Paticle and the auxiliay equations ae pesented below: G N T N f ma a v 0 G3 Mg f f mg N N mg (6) By solving the component's equation set of the above-stated vecto equation set, the following is obtained: T3 f mg N mg N3 ( m M ) g (7) In the pocess when the foce-applying Paticle moves along the positive diection of xis x in the distance s, the wok done by foces on Paticle is: The wok by the consevative foces x0 0 c c3 x0 Then calculate the wok of Paticle by non-consevative foces Calculate the implicated wok by non-consevative foces G3 d (8) N d N dx dx 0 si x0 s x0 s nci () cos( ) x 0 x (9) 0 0 si x0 s nci() cos0 ( ) 0 x (30) 0 f d f dx f s mgs 37 Volume 6, Numbe, pil 06

8 Intenational Jounal of pplied Physics and Mathematics N3 d3 (3) 30 nci3() 0 30 The foce-applying object, i.e. the gound, is the efeence and theefoe has no absolute displacement. Refe to.3... T3 d3 (3) 30 nci3() 0 30 Substitute Equations (9)-(3) into the equation below, to obtain the total implicated wok of Paticle done by non-consevative foces, as below: 3 nci ncij j, nci() nci() nci3() nci3() f ( ) s mgs (33) Calculate the dissipated wok by non-consevative foces N d N dx dx 0 si x0 s x0 s ncd () cos( ) x 0 x (34) 0 f d f dx mgs 0 si x0 s ncd () cos0 0 x (35) ncd 3() N d 3 3 (36) ncd 3() T d 3 3 (37) 30 Substitute Equations (34)-(37) into the equation below, to obtain the total dissipated wok of Paticle done by non-consevative foces, as below: 3 ncd ncd j j, ncd () ncd () ncd 3() ncd 3() mgs (38) Calculate the dissipated wok by non-consevative eacting foces The following is obtained fom Equation (9): ncd ncd mgs (39) 38 Volume 6, Numbe, pil 06

9 Intenational Jounal of pplied Physics and Mathematics Calculate the incement of kinetic enegy in Paticle When Paticle is still: Use equations to solve E E M k k0 0 0 Ek Ek Ek 0 0 (40) Substitute Equations (8), (33), (38), (39) and (40) into the Equation () - the NTKE fo paticles - to obtain: f s ( μmgs) ( μmgs) 0 (4) The ight pat to the equation's equal sign indicates: in the moving pocess of the foce-applying object, the incement of kinetic enegy in Paticle which is still and unde foce is zeo. The left pat to the equation's equal sign indicates: total mechanical enegy (value: f s μmgs ) of Paticle conveted fom the non-dissipated wok of Paticle by pull, half of which, due to the dissipated wok (the nd tem) done by non-consevative foces, is conveted into the non-mechanical enegy (value: μmgs ) of the foce-beaing paticle, and anothe half, due to the dissipated wok (the 3d tem) of the foce-applying paticle by the non-consevative foces, is conveted into the non-mechanical enegy [4] (value: μmgs ) of the foce-beaing paticle. Objective chaacteization of the tansfeing and convesion of the still Paticle that is acted by sliding fiction foce cannot be pesented with the classical kinetic enegy theoem because in tems of CTKE, the wok-enegy equation fo the still Paticle is: 0=0 (4) It is impossible to get data about tansfeing and convesion of mechanical enegy in the still Paticle that is unde the sliding fiction foce using the Equation (4). Theefoe, wheeve any non-consevative foce is pesent, the NTKE on paticle, athe than the CTKE, shall be used, because the latte is applicable in the case that thee exists no consevative foce..3. New Theoem of Kinetic Enegy on Paticle System The NTKE on paticle system is expessed as: in the motion pocess of a system, the sum of woks (infinitesimal) of a paticle system by all consevative foces, the dissipated wok all non-consevative foces, the dissipated wok foces, and the implicated wok incement (infinitesimal) of total kinetic enegy in the system. The diffeential expession is: c ncd (infinitesimal) by ncd (infinitesimal) by all non-consevative eacting ex nci (infinitesimal) by all non-consevative extenal foces equals to the The integal expession is: ex d c +d ncd +d ncd +d nci = de k (43) 39 Volume 6, Numbe, pil 06

10 Intenational Jounal of pplied Physics and Mathematics = E - E (44) ex c ncd ncd nci k k0 whee, = + ( c c c d = d +d ) (45) c c c = + ( d = d +d ) (46) ncd ncd ncd ncd ncd ncd = + ( d = d +d ) (47) ncd ncd ncd ncd ncd ncd In the above-pesented equations, tems with the pe-supescipt "in" ae woks associated with the intenal foces within the system while the tems with the pe-supescipt "ex" ae woks coesponding to the extenal foces. Thee is no total implicated wok by non-consevative intenal foces, indicating that the implicated wok associated with the non-consevative intenal foces ae the values of mechanical enegy tansfeed between paticles within the system, which does not change the total kinetic enegy in the system..4. Exploation on pplication of NTKE on Paticle System Example : as shown in Fig.3, two objects and B in espective mass of m and m ae hung on both sides of the light pulley. is placed ove the slope at the angle θ and the fiction facto between and the slope is. Solve the woks done by all foces and the end speeds of two paticles when B acceleates downwads stating with stillness by height of Δh? (assuming the elongation of the light ope is negligible and the axis of the light pulley is smooth). Fig. 3. Paticle goud motion. nswe: use the NTKE on paticle system to solve. Using the slope (the Eath) as the efeence, set up a 0-xy coodinate system. Because both and B moves in linea way, both can be teated as paticle. Paticles and B constitute a system. Define the numbe of objects inteacting against each othe: : Paticle ; : Paticles B; 3: the light ope; 4: the slope (the Eath). ) nalyze the foces bone by membes of the system Foces bone by Paticle at the time t when it moves to the position ( x,0 ): Thee is no intenal foce between Paticles and B. System's extenal foces: The pull fom the light ope: f 3 ; the gavity F4() mg ; the suppot foce of the slope: f 4() and the f4(3) slope fiction foce:. The non-consevative extenal eacting foces include: 40 Volume 6, Numbe, pil 06

11 Intenational Jounal of pplied Physics and Mathematics The pull of Paticle elative to the light ope: f 3; the pessue against the slope: f 4() ; and the fiction foce against the slope: f 4(3). Extenal foces bone by the system at the time t when Paticle B moves to the position ( x, y): The gavity F4 mg and the pull of the light ope f 3. The non-consevative extenal eacting foces include: The pull of Paticle B elative to the light ope f 3 ) Since Paticles and B do acceleating linea motion espectively, the component equations fo Newton's Second Law and the auxiliay equation sets ae pesented below: Fx f f F sin m a Patcile : Fy f4() F4() cos 0 f4(3) f4(3) μf4() F4() m g F4 f3 ma PatcileB : F4 mg a a f4() f4() f3 f3 f3 f3 3 4(3) 4() (48) By solving the above-pesented equation set, the following is obtained: m m(sinθ cos ) a a g m m mm g f3 f3 f3 f3 ( sinθ cos ) m m f4() f4() m g cos f4(3) f4(3) m g cos (49) 3) Because both paticles do linea motions espectively and all foces bone ae constant, the method to calculate the wok by constant-foces, can be used, in combination with the new definition equation of espective woks, to calculate. While Paticle B stats fom stillness and acceleates downwads by height of Δh,Paticle stats fom stillness to acceleate in linea way along the slope at the displacement of Δx=Δh. Based on the Equation (38) fo NTKE on paticle system, the woks ae calculated: Total wok of the system by the consevative foces: = + (50) c c c The intenal foce is nonexistent: 4 Volume 6, Numbe, pil 06

12 Intenational Jounal of pplied Physics and Mathematics 0 (5) in c ex ex ex c c4() c4 m g sinx m gh ( m m sin ) gh (5) Substitute Equations (5) and (5) into (45), we obtains: ( m m sin ) g h. (53) c The consevative extenal foces do non-dissipated wok to the paticle system to tansfe the mechanical enegy to the paticle system. Calculate total dissipated wok of the system by the consevative foces: = +. (54) ncd ncd ncd The intenal foce is nonexistent: Total dissipated wok of the system by the non-consevative foces: Tems with known value: 0 (55) in ncd No elative displacement of the paticle's foce-beaing point elative to the ope's foce-applying point. x (56) 3 x3 0 x h (57) 4 Substitute the above-mentioned known values into the fomula below: = + ex ex ex ncd ncd ncd ( ) ex ex ex ex ncd3 ncd4() ncd4(3) ncd 3 ( f3 x3 f4() x4 f4(3) x4 f3 x3 [ f f x f x f m gh cos 3 4() 4 4(3) 4 3 (58) Non-consevative extenal foces do dissipated wok to the paticle system so that pat of mechanical enegy tansfeed to the paticle system is conveted into the non-mechanical enegy in the paticle system. Substitute Equations (49) and (5) into (48) to obtain: 4 Volume 6, Numbe, pil 06

13 Intenational Jounal of pplied Physics and Mathematics ncd m g hcos (59) Calculate total dissipated wok of the system by the non-consevative eacting foces: Fom Equation (9), the following is obtained: ncd ncd m g hcos (60) Calculate total implicated wok of the system by the non-consevative foces: Tems with known value: The implicated displacement of the Paticle 's foce-applying point by the light ope equals to the displacement of Paticle 's foce-beaing point: Using the slope as efeence, thee is no implicated displacement: x3 x h. (6) x 4 0. (6) The implicated displacement of the Paticle B's foce-applying point by the light ope equals to the displacement of Paticle B's foce-beaing point: Substitute (6), (6) and (63) into the equation below: 3 x h. (63) ( ) ex ex ex ex ex nci nci3 nci4() nci4(3) nci3 ( f x f x f x f 3 3 4() 4 4(3) ( f x cos0 f f f () 4(3) 3 3 (64) Calculate the incement of kinetic enegy of paticle system fo the whole jouney v v 0 v v 0 0 (65) E E ( m v ) ( m v ) k k 0 i i i i0 i i ( mv mv ) 0 ( m m) v (66) Get equations to wok out the unknown quantities 43 Volume 6, Numbe, pil 06

14 Intenational Jounal of pplied Physics and Mathematics Substitute Equations (53), (59), (60), (64) and (66) into the paticle system NTKE equation (44) to obtain: ( m ( m m) v m sin ) gh m gh cos m gh cos 0 ( m m) v (67) The esult is: ( m - m sinθ- μm cosθ) v v gδ m m h (68) Discussion: Only when m - msin θ- μm cosθ 0, the paticle system can do the motion in the above-assumed condition. 3. Conclusion The NTKE on paticle system is deived fom that on paticles. NTKE, whethe on paticles o on paticle system, can eveal actual tansfeing and convesion of mechanical enegy and necessaily povide coect guidance to pactices. The NTKE shall be flexibly used on the basis of pactical conditions and equiements to set and solve equations. Refeence: [] Yang,Y. (00). Study on new wok-enegy theoy. New Couse Leaning, 33, [] Yang, Y. (04). Discussion on paticle's absolute, elative and implicated motions. Compilation of Scientific Educational Papes,, [3] Zhou, Y. B. (979). Couse on Theoetical Mechanics, 9-3. Beijing: Highe Education Pess. [4] Yang, Y. (00). Discussion on the deductive theoies of system wok-enegy pinciple. cademic Jounal of Shaanxi Nomal Univesity, 30, 4-6. [5] Yang, Y. (005). Demonstation and dissemination of deduction fom system wok-enegy theoy. Jounal of Shaanxi Nomal Univesity (fo Natual Science), 33, -5, 9. [6] Yang, Y. (009). cting-foce does wok, so does the eacting foce: utomation of manufactue,, 6-8. Yang Yong was bon in Tongchuan city, Sha anxi povince, China, in Novembe 959. He is an associate pofesso in physics and a contact eseach fellow of both Cente fo Futue Studies, China Society fo Futue Studies, and of Chinese cademy of Management Science. He gaduated fom the Depatment of Physics with Sha anxi Nomal Univesity in July 983 fo the bachelo degee and fom Gaduate School of Sha anxi Nomal Univesity majoing in cuiculum and teaching theoy in 004 fo maste degee. He had been teaching physics at the Depatment of Mathematics and Physics in Tongchuan College of Education and Tongchuan Vocational and Technical College fo 34 yeas. With eseach focused on Dynamics of Paticles, he had published papes such as on a deivation fom the theoy of wok-enegy of systems and study on new theoy of wok-enegy. He evealed theoetically and discoveed non-consevative foce does dissipated wok, so does its eacting foce and non-consevative foce also does non-dissipated wok fistly, and then by futhe deivation came up with the new theoy of wok-enegy. In this way, he unified the theoy of wok-enegy and pactices at a new level and thus developed the classical theoy on wok-enegy into a new couse. 44 Volume 6, Numbe, pil 06