本教材僅供教學使用, 勿做其他用途, 以維護智慧財產權

Transcription

1 本教材內容主要取自課本 Physcs fo Scentsts and Engnees wth Moden Physcs 7th Edton. Jewett & Seway. 注意 本教材僅供教學使用, 勿做其他用途, 以維護智慧財產權 教材網址 : 1

2 Chapte 12 Statc Equlbum and Elastcty 2

3 女王頭, 台北縣野柳地質公園 Balanced Rock n Aches Natonal Pak, Utah, s a kg boulde that has been n stable equlbum fo seveal mllenna. It had a smalle companon neaby, called Chp Off the Old Block, that fell dung the wnte of Balanced Rock appeaed n an ealy scene of the move Indana Jones and the Last Cusade. We wll study the condtons unde whch an object s n equlbum n ths chapte. 3

4 The Rgd Object n Equlbum Secton

5 Statc Equlbum Equlbum mples the object s at est (statc equlbum) o ts cente of mass moves wth a constant velocty (dynamc equlbum) Statc equlbum s a common stuaton n engneeng Pncples nvolved ae of patcula nteest to cvl engnees, achtects, and mechancal engnees 5

6 Statc vs. Dynamc Equlbum In ths chapte, we wll concentate on statc equlbum The object wll not be movng v CM = 0 and ω = 0 a CM = 0, α =0 ΣF = 0, Στ= 0 Howeve, when & a CM = 0, α = 0 Dynamc equlbum s also possble The object would be otatng wth a constant angula velocty ω The object would be movng wth a constant v CM 6

7 Condtons fo Equlbum The net foce equals zeo If the object s modeled as a patcle, then ths s the only condton that must be satsfed The net toque equals zeo a CM = 0 α= 0 Ths s needed f the object cannot be modeled as a patcle These condtons descbe the gd objects n equlbum analyss model 7

8 Tanslatonal Equlbum The fst condton of equlbum s a statement of tanslatonal equlbum a CM = 0 It states that the tanslatonal acceleaton of the object s cente of mass must be zeo Ths apples when vewed fom an netal efeence fame The object would be movng wth a constant v CM 8

9 Rotatonal Equlbum The second condton of equlbum s a statement of otatonal equlbum α= 0 It states the angula acceleaton of the object to be zeo The object would be otatng wth a constant angula velocty ω Ths must be tue fo any axs of otaton 9

10 10 O O m F = 0 = F τ 0 ) ( = = = = = F F F F F τ = 0 F 0 0 = 0 τ Equlbum 靜力平衡時, 若對某一轉軸 O 計算所得的力矩為零, 則對任意轉軸 O 亦可得到力矩為零的結果

11 Equlbum Equatons & 6 equatons We wll estct the applcatons to stuatons n whch all the foces le n the xy plane F z = 0, τ x = 0, τ y = 0 leave 3 equatons These ae called coplana foces snce they le n the same plane Thee ae thee esultng equatons ΣF x = 0 ΣF y = 0 Στ = 0 11

12 Axs of Rotaton fo Toque Equaton The net toque s about an axs though any pont n the xy plane The choce of an axs s abtay If an object s n tanslatonal equlbum and the net toque s zeo about one axs, then the net toque must be zeo about any othe axs 12

13 Toque = τ F Use the ght hand ule to detemne the decton of the toque The tendency of the foce to cause a otaton about O depends on F and the moment am d 13

14 Answe: 14

15 Answe: 15

16 Moe on the Cente of Gavty Secton

17 Cente of Mass An object can be dvded nto many small patcles Each patcle wll have a specfc mass and specfc coodnates The x coodnate of the cente of mass wll be Smla expessons can be found fo the y and z coodnates 17

18 Cente of Gavty All the vaous gavtatonal foces actng on all the vaous mass elements ae equvalent to a sngle gavtatonal foce actng though a sngle pont called the cente of gavty (CG) τ = x m g f GC g τ = Thus = 1 M = g x ( ) = const. m m x g = = CM GC Mg 18

19 Cente of Gavty, cont The toque due to the gavtatonal foce on an object of mass M s the foce Mg actng at the cente of gavty of the object If g s unfom ove the object, then the cente of gavty of the object concdes wth ts cente of mass If the object s homogeneous and symmetcal, the cente of gavty concdes wth ts geometc cente Howeve, f = x τ g const. ( m g ) = GC MgGC 19

20 Examples of Rgd Object n Statc Equlbum Secton

21 Fg. 12-7, p. 341 Ths one-bottle wne holde s a supsng dsplay of statc equlbum. The cente of gavty of the system (bottle plus holde) s dectly ove the suppot pont. 21

22 22

23 23

24 24

25 Hozontal Beam Example The beam s unfom So the cente of gavty s at the geometc cente of the beam The peson s standng on the beam What ae the tenson n the cable and the foce exeted by the wall on the beam? 25

26 Hozontal Beam Example, 2 Analyze Daw a fee body dagam Use the pvot n the poblem (at the wall) as the pvot Ths wll geneally be easest Note thee ae thee unknowns (T, R, θ) 26

27 Hozontal Beam Example, 3 The foces can be esolved nto components n the fee body dagam Apply the two condtons of equlbum to obtan thee equatons Solve fo the unknowns 27

28 28

29 29

30 30

31 Ladde Example The ladde s unfom So the weght of the ladde acts though ts geometc cente (ts cente of gavty) Thee s statc fcton between the ladde and the gound 31

32 Ladde Example, 2 Analyze Daw a fee body dagam fo the ladde The fctonal foce s ƒ s = µ s n Let O be the axs of otaton Apply the equatons fo the two condtons of equlbum Solve the equatons 32

33 33

34 34

35 35

36 36

37 37

38 38

39 Elastc Popetes of Solds Secton

40 Elastcty So fa we have assumed that objects eman gd when extenal foces act on them Except spngs Actually, objects ae defomable It s possble to change the sze and/o shape of the object by applyng extenal foces Intenal foces esst the defomaton 40

41 Defntons Assocated Wth Defomaton Stess ( 應力 ) Is popotonal to the foce causng the defomaton It s the extenal foce actng on the object pe unt aea (ex. F/A) Stan ( 形變 ) Is the esult of a stess Is a measue of the degee of defomaton (ex. L/L ) 41

42 Elastc Modulus The elastc modulus s the constant of popotonalty between the stess and the stan Fo suffcently small stesses, the stan s dectly popotonal to the stess (ex. stess = Y stan) It depends on the mateal beng defomed It also depends on the natue of the defomaton 42

43 Elastc Modulus, cont The elastc modulus, n geneal, elates what s done to a sold object to how that object esponds stess elastc modulus = k = F/x stan Vaous types of defomaton have unque elastc modul 43

44 Thee Types of Modul Young s Modulus Measues the esstance of a sold to a change n ts length Shea Modulus Measues the esstance of moton of the planes wthn a sold paallel to each othe Bulk Modulus Measues the esstance of solds o lquds to changes n the volume 44

45 Young s Modulus The ba s stetched by an amount L unde the acton of the foce F See the actve fgue fo vaatons n values Stan = L / L The tensle stess s the ato of the magntude of the extenal foce to the coss-sectonal aea A stess = F / A 45

46 Young s Modulus, cont The tenson stan s the ato of the change n length to the ognal length Young s modulus, Y, s the ato of those two atos: Unts ae N / m 2 46

47 Stess vs. Stan Cuve Expements show that fo cetan stesses, the stess s dectly popotonal to the stan Ths s the elastc behavo pat of the cuve pemanently defomed 47

48 Stess vs. Stan Cuve, cont The elastc lmt s the maxmum stess that can be appled to the substance befoe t becomes pemanently defomed When the stess exceeds the elastc lmt, the substance wll be pemanently defomed The cuve s no longe a staght lne Wth addtonal stess, the mateal ultmately beaks 48

49 Shea Modulus Anothe type of defomaton occus when a foce acts paallel to one of ts faces whle the opposte face s held fxed by anothe foce See the actve fgue to vay the values Ths s called a shea stess (F/A) 49

50 Fg b, p. 349 A book s unde shea stess when a hand placed on the cove apples a hozontal foce away fom the spne. 50

51 Shea Modulus, cont Fo small defomatons, no change n volume occus wth ths defomaton A good fst appoxmaton The shea stess s F / A F s the tangental foce A s the aea of the face beng sheaed The shea stan s x / h x s the hozontal dstance the sheaed face moves h s the heght of the object 51

52 Shea Modulus, fnal The shea modulus s the ato of the shea stess to the shea stan Unts ae N / m 2 52

53 Plastc Behavo Stess, Stan, and Elastcty σ: 張應力 tensle stess ε: 形變 stan Y : 楊氏係數 Young s modulus x [ σ ν ( σ σ )] Y ε = + x ν : Posson s ato ~ 0.3 y z τ xy : x 面受 y 方向剪力 shea stess γ : 剪力形變 shea stan µ : shea modulus Fo sotopc mateals: µ=y/2(1+ν) 53

54 Bulk Modulus Anothe type of defomaton occus when a foce of unfom magntude s appled pependculaly ove the ente suface of the object See the actve fgue to vay the values The object wll undego a change n volume, but not n shape 54

55 The volume stess s defned as the ato of the magntude of the total foce, F, exeted on the suface to the aea, A, of the suface Ths s also called the pessue (F/A) The volume stan s the ato of the change n volume to the ognal volume ( V / V ) The bulk modulus (B) s the ato of the volume stess to the volume stan The negatve ndcates that an ncease n pessue wll esult n a decease n volume 55

56 Compessblty ( 可壓縮性 ) The compessblty s the nvese of the bulk modulus compessblty = 1/B It may be used nstead of the bulk modulus 56

57 Modul and Types of Mateals Both solds and lquds have a bulk modulus Lquds cannot sustan a sheang stess o a tensle stess If a sheang foce o a tensle foce s appled to a lqud, the lqud wll flow n esponse 57

58 Modul Values 58

59 59

60 60