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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 : https://stes.google.com/ste/ndhugp1/ 1

Chapte 12 Statc Equlbum and Elastcty 2

女王頭, 台北縣野柳地質公園 Balanced Rock n Aches Natonal Pak, Utah, s a 3 000 000-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 1975. 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

The Rgd Object n Equlbum Secton 12.1 4

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

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

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

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

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 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

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

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

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

Answe: 14

Answe: 15

Moe on the Cente of Gavty Secton 12.2 16

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

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

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

Examples of Rgd Object n Statc Equlbum Secton 12.3 20

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

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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

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

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

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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

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

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Elastc Popetes of Solds Secton 12.4 39

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

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

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

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

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

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

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

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

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

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

Fg. 12-14b, p. 349 A book s unde shea stess when a hand placed on the cove apples a hozontal foce away fom the spne. 50

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

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

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

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

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

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

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

Modul Values 58

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