Chapte 12 Equlbum and Elastcty In ths chapte we wll defne equlbum and fnd the condtons needed so that an object s at equlbum. We wll then apply these condtons to a vaety of pactcal engneeng poblems of statc equlbum. We wll also examne how a gd body can be defomed by an extenal foce. In ths secton we wll ntoduce the followng concepts: Stess and stan Young s modulus (n connecton wth tenson and compesson) Shea modulus (n connecton wth sheang) Bulk modulus (n connecton wth hydaulc stess) (12-1)
Equlbum We say that an object s n equlbum when the followng two condtons ae satsfed: 1. The lnea momentum P of the cente of mass s constant. 2. The angula momentum L about the cente of mass o any othe pont s a constant. Ou concen n ths chapte s wth stuatons n whch P = 0 and L = 0. That s, we ae nteested n objects that ae not movng n any way (ths ncludes tanslatonal as well as otatonal moton) n the efeence fame fom whch we obseve them. Such objects ae sad to be n statc equlbum. In Chapte 8 we dffeentated between stable and unstable statc equlbum. If a body that s n statc equlbum s dsplaced slghtly fom ths poston the foces on t may etun t to ts old poston. In ths case we say that the equlbum s equlbum s stable. If the body does not etun to ts old poston then the unstable. (12-2)
An example of unstable equlbum s shown n the fgues. In fg. a we balance a domno wth the domno's cente of mass vetcally above the suppotng edge. The toque of the gavtatonal foce g about the suppotng edge s zeo because the lne of acton of passes though the edge. g Thus the domno s n equlbum. Even a slght foce on the domno ends the equlbum. As the lne of acton of g moves to one sde of the suppotng edge (see fg. b) the toque due to s nonzeo and the domno otates n the clockwse decton away fom ts equlbum poston of fg. a. The domno n fg. a s n a poston of unstable equlbum. The domno s fg. c s not qute as unstable. To topple the domno the appled foce would have to otate t though and beyond the poston of fg. a. A flck of the fnge aganst the domno can topple t. g (12-3)
The Condtons of Equlbum In Chapte 9 we calculated the ate of change fo the lnea momentum of the cente dp of mass of an object, net. If an object s n tanslatonal equlbum the dt = n dp P = constant and thus = 0 net = 0. dt In Chapte 11 we analyzed otatonal moton and saw that Newton's second law takes the dl fom = net. o an object n otatonal equlbum we have: L = constant dt dl = 0 net = 0. dt The two equements fo a body to be n equlbum ae: 1. The vecto sum of all the extenal foces on the body must be zeo. 2. The vecto sum of all the extenal toques that act on the body measued about net = 0 net = 0 any pont must be zeo. (12-4)
In component fom the condtons of equlbum ae: Balance of foces: = 0 = 0 = 0 net, x net, y net, z Balance of toques: = 0 = 0 = 0 net, z net, x net, y net, z We shall smplfy mattes by consdeng only poblems n whch all the foces that act on the body le n the xy-plane. Ths means that the only toques geneated by these foces tend to cause otaton about an axs paallel to the z-axs. Wth ths assumpton the condtons fo equlbum become: Balance of foces: Balance of toques: net, x net, z = 0 = 0 = 0 net, y Hee s the net toque poduced by all extenal foces ethe about the z-axs o about any axs paallel to t. nally, fo statc equlbum the lnea momentum P of the cente of mass must be zeo: P = 0. net, x net, z = 0 = 0 = 0 net, y (12-5)
Statcs Poblem Recpe 1. Daw a foce dagam. (Label the axes.) 2. Choose a convenent ogn O. A good choce s to have one of the unknown foces actng at O. 3. Sgn of the toque fo each foce: - If the foce nduces clockwse (CW) otaton + If the foce nduces counteclockwse (CCW) otaton 4. Equlbum condtons: 5. Make sue that numbe of unknowns = numbe of equatons (12-6) net, x net, z = 0 = 0 = 0 net, y
The Cente of Gavty (cog) The gavtatonal foce actng on an extended body s the vecto sum of the gavtatonal foces actng on the ndvdual elements of the body. The gavtatonal foce on a body effectvely acts at a sngle pont known as the cente of gavty g of the body. Hee "effectvely" has the followng meanng: If the ndvdual gavtatonal foces on the elements of the body ae tuned off and eplaced by g actng at the cente of gavty, then the net foce and the net toque about any pont on the body do not change. We shall pove that f the acceleaton of gavty g s the same fo all the elements of the body then the cente of gavty concdes wth the cente of mass. Ths s a easonable appoxmaton fo objects nea the suface of the Eath because g changes vey lttle. (12-7)
Consde the extended object of mass M shown n fg. a. In fg. a we also show the th element of mass m. The gavtatonal foce on m s equal to mg whee g s the acceleaton of gavty n the net vcnty of m. The toque on m s equal to x. The net toque = = (eq. 1). Consde now fg. b n whch we have eplaced the foces by the net gavtatonal foce g x g actng at the cente of gavty. The net g net net = cog g = cog toque s equal to: x x (eq. 2). g g If we compae equaton 1 wth equaton 2 we get: x = x. We substtute mg fo and we have: x mg = mgx. g cog mx If we set g = g fo all the elements xcog = m = xcom. cog g g (12-8)
O Sample Poblem 12-1. A unfom beam of length L and mass m = 1.8 kg s at est on two scales. A unfom block of mass M = 2.7 kg s at est on the beam at a dstance L/4 fom ts left end. Calclat u e the scales eadngs: = + Mg mg = 0 (eq. 1) net, y l We choose to calculate the toque wth espect to an axs though the left end of the beam (pont O). L L net, z = ( mg ) ( Mg ) + ( L)( ) = 0 (eq. 2) 4 2 Mg mg 2.7 9.8 1.8 9.8 om equaton 2 we get: = + = + = 15.44 15 N. 4 2 4 2 We solve equaton 1 fo = Mg+ mg = 2.7+1.8 9.8 15.44 = 28.66 N: l 29 N. l l ( ) (12-9)
We take toques about an axs though pont O. net, x w px px w net, y py py Sample Poblem 12-2. A ladde of length L = 12 m and mass m = 45 kg leans aganst a fctonless wall. The ladde's uppe end s at a heght h = 9.3 m above the pavement on whch the lowe end ests. The com of the ladde s L/3 fom the lowe end. A fefghte of mass M = 72 kg clmbs half way up the ladde. nd the foces exeted on the ladde by the wall 2 2 and the pavement. Dstance a L h 7.5 a a net, z = ( h)( w) + ( mg) + ( Mg) = 0 3 2 M m ga + 2 3 9.8 7.58 ( 72 / 2 + 45 / 3) w = = = 407 N 410 N h 9.3 = = 0 = = 410 N ( ) = = 8 m. = Mg mg = 0 = Mg+ mg = 9.8 72+ 45 = 1146.6 N 1100 N (12-10)
net, x h c h c ( ) ( ) Sample Poblem 12-3. A safe of mass M = 430 kg hangs by a ope fom a boom wth dmensons a = 1.9 m and b = 2.5 m. The beam of the boom has mass m = 85 kg. nd the tenson = T = 0 = T = 6093 N net, y v v 2 2 h v T c n the cable and th e magntude of the net foce exeted on the beam by the hnge. We calculate the net toque about an axs nomal to the page that passes though pont O. b net, z = ( a)( Tc) ( b)( T) ( mg) = 0 2 m gb M + 2 9. T 8 2.5( 430+ 85/2) c = = 6100 N a 1.9 2 2 ( ) ( ) = mg T = 0 = mg + T = g m + M = 9.8 85 + 430 = 5047 N = + = 6093 + 5047 7900 N (12-11)
O Sample Poblem 12-4. A 70 kg ock clmbe hangs by the cmp hold of one hand. He feet touch the ock dectly below he fnges. Assume that the foce fom the hozontal ledge suppotng he fnges s equally shaed by the fou fnges. Calculate the hozontal and vetcal components and of the foce on each fngetp. = + 4 = 0 net, x N h mg 70 9.8 net, y = 4v mg = 0 v = = = 171.5 N 170 N 4 4 h v We calculate the net toque about an axs that s pependcula to the page and passes though pont O. ( ) ( )( mg) ( )( ) ( )( ) = 0 + 0.2 2.0 4 + 0 4 = 0 net, z N h v h 0.20 70 9.8 = = 17.15 N 17 N 4 2.0 (12-12)
Indetemnate Stuctues o the poblems n ths chapte we have the followng thee equatons at ou dsposal: = 0 = 0 = 0 net, x net, y net, z If the poblem has moe than thee unknowns we cannot solve t. We can solve a statcs poblem fo a table wth thee legs but not fo one wth fou legs. Poblems lke these ae called ndetemnate. An example s gven n the fgue. A bg elephant sts on a wobbly table. If the table does not collapse t wll defom so that all fou legs touch the floo. The upwad foces exeted on the legs by the floo assume defnte and dffeent values. How can we calculate the values of these foces? To solve such an ndetemnate equlbum poblem we must supplement the thee equlbum equatons wth some knowledge of elastcty, the banch of physcs and engneeng that descbes how eal bodes defom when foces ae appled to them. (12-13)
Elastcty Metallc solds consst of a lage numbe of atoms postoned on a egula thee-dmensonal lattce as shown n the fgue. The lattce s epetton of a patten (n the fgue ths patten s a cube). Each atom of the sold s a well-defned equlbum dstance fom ts neaest neghbos. The atoms ae held togethe by nteatomc foces that can be modeled as tny spngs. If we ty to change the nteatomc dstance the esultng foce s popotonal to the atom dsplacement fom the equlbum poston. The spng constants ae lage and thus the lattce s emakably gd. Nevetheless all "gd" bodes ae to some extent elastc, whch means that we can change the dmensons slghtly by pullng, pushng, twstng, o compessng them. o example, f you suspend a subcompact ca fom a steel od 1 m long and 1 cm n damete, the od wll stetch by only 0.5 mm. The od wll etun to ts ognal length of 1 m when the ca s emoved. If you suspend two cas fom the od the od wll be pemanently defomed. If you suspend thee cas the od wll beak. (12-14)
stess = modulus stan In the thee fgues above we show the thee ways n whch a sold mght change ts dmensons unde the acton of extenal defomng foces. In fg. a the cylnde s stetched by foces actng along the cylnde axs. In fg. b the cylnde s defomed by foces pependcula to ts axs. In fg. c a sold placed n a flud unde hgh pessue s compessed unfomly on all sdes. All thee defomaton types have stess n common (defned as defomng foce pe unt aea). These stesses ae known as tensle/compessve fo fg. a, sheang fo fg. b, and hydaulc fo fg. c. The applcaton of stess on a sold esults n stan, whch takes dffeent fom fo the thee types of stan. Stan s elated to stan va the equaton : stess = modulus stan. (12-15)
= E A A ΔL L Tensle stess s defned as the ato whee A s the sold aea. A ΔL Sta n (symbol S) s defned as the ato whee ΔL L s the change n the length L of the cylndcal sold. Stess s plotted vesus stan n the uppe fgue. o a wde ange of appled stesses the stess-stan elaton s lnea and the sold etuns to ts ognal length when the stess s emoved. Ths s known as the elastc ange. If the stess s nceased beyond a maxmum value known as the yeld stength the cylnde becomes pemanently defomed. If the stess contnues to ncease, the at a stess value known as ultmate stength S u. S y cylnde beaks o stesses below (elastc ange) stess and stan ae connected va the equaton S y ΔL = E. The c onstant E (modulus) s known as Young's modulus. A L Note: Young's modulus s almost the same fo tenson and compesson. The ultmate stength S u may be dffeent. (12-16)
A A A Δx = G L ΔV p = B V Shea ng. In the case of sheang defomaton, stan s defned as the dmensonless ato equaton has the fom: Δx L Δx = G A L The constant G s known as the shea modulus.. The stess/stan Hydaulc Stess. The stess s ths case s the pessue p = that A the suoundng flud exets on the mmesed object. Hee A s the aea of the object. In ths case stan s defned as the ΔV dmensonless ato whee V s the volume of the object V and ΔV the change n the volume due to the flud pessue. The ΔV stess/stan equaton has the fom: p = B. The constant V B s known as th e bulk mod ulus of the mateal. (12-17)