PHYS 1443 Section 002 Lecture #20

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1 PHYS 1443 Secton 002 Lecture #20 Dr. Jae Condtons for Equlbru & Mechancal Equlbru How to Solve Equlbru Probles? A ew Exaples of Mechancal Equlbru Elastc Propertes of Solds Densty and Specfc Gravty lud and Pressure Pascal s Prncple Today s hoework s None!! PHYS , all 2008 Dr. Jaehoon 1

2 Announceents 3 rd ter exa results s Average: 48.3/94 Equvalent to 51.4/100 Prevous exas: 62/100 and 67/100 Top score: 73/94 nal exa Date and Te: 11a 12:30p, next Monday, Dec. 8 Locaton: SH103 Coprehensve exa: Covers CH What we fnsh ths Wednesday, Dec. 3 + appendces Mxture of ultple choce and free response probles PHYS , all 2008 Dr. Jaehoon 2

3 Condtons for Equlbru What do you thnk the ter An object s at ts equlbru eans? The object s ether at rest (Statc Equlbru) or ts center of ass s ovng at a constant velocty (Dynac Equlbru). When do you thnk an object s at ts equlbru? Translatonal Equlbru: Equlbru n lnear oton 0 Is ths t? d d CM - The above condton s suffcent for a pont-lke object to be at ts translatonal equlbru. or an object wth sze, however, ths s not suffcent. One ore condton s needed. What s t? Let s consder two forces equal n agntude but n opposte drecton actng on a rgd object as shown n the fgure. What do you thnk wll happen? The object wll rotate about the CM. The net torque actng on the object about any axs ust be 0. τ 0 or an object to be at ts statc equlbru, the object should not have lnear or angular speed. v 0 ω 0 PHYS , all 2008 Dr. Jaehoon CM 3

4 More on Condtons for Equlbru To splfy the proble, we wll only deal wth forces actng on x-y plane, gvng torque only along z-axs. What do you thnk the condtons for equlbru be n ths case? The sx possble equatons fro the two vector equatons turns to three equatons. 0 x 0 AND τ 0 y 0 τ z 0 What happens f there are any forces exertng on an object? r O r 5 O If an object s at ts translatonal statc equlbru, and f the net torque actng on the object s 0 about one axs, the net torque ust be 0 about any arbtrary axs. Why s ths true? Because the object s not ovng, no atter what the rotatonal axs s, there should not be any oton. It s sply a atter of atheatcal anpulaton. PHYS , all 2008 Dr. Jaehoon 4

5 Center of Gravty Revsted When s the center of gravty of a rgd d body the sae as the center of ass? Under the unfor gravtatonal feld throughout the body of the object. CM CoG ( 1g1 + 2g 2 + ) x CoG Let s consder an arbtrary shaped object The center of ass of ths object s at x CM y CM x Let s now exane the case that the gravtatonal acceleraton on each pont s g y Snce the CoG s the pont as f all the gravtatonal force s exerted on, the torque due to ths force becoes 1g1x1 + 2g 2x2 + x M M Generalzed expresson for dfferent g throughout the body y If g s unfor throughout the body ( ) gxcog 1 ( x + x + )g PHYS , all 2008 Dr. Jaehoon x CoG x x CM 5

6 How do we solve equlbru probles? 1. Identfy all the forces and ther drectons and locatons 2. Draw a free-body dagra wth forces ndcated on t wth ther drectons and locatons properly noted 3. Wrte down force equaton for each x and y coponent wth proper sgns 4. Select a rotatonal axs for torque calculatons Selectng the axs such that t the torque of one of the unknown forces becoe 0 akes the proble easer to solve 5. Wrte down the torque equaton wth proper sgns 6. Solve the equatons for unknown quanttes PHYS , all 2008 Dr. Jaehoon 6

7 Exaple for Mechancal Equlbru A unfor N board supports the father and the daughter each weghng 800 N and 350 N, respectvely, and s not ovng. If the support (or fulcru) s under the center of gravty of the board, and the father s 1.00 fro CoG, what s the agntude of the noral force n exerted on the board by the support? M g 1 n M B g x D M D g Therefore the agntude of the noral force Snce there s no lnear oton, ths syste s n ts translatonal equlbru 0 x y n n Deterne where the chld should st to balance the syste. M B g M g M g N The net torque about the fulcru τ M by the three forces are B g 0 + n 0 + M g 1.00 M D Therefore to balance the syste x M g the daughter ust st M D g 350 PHYS , all 2008 Dr. Jaehoon D g x 0 7

8 Exaple for Mech. Equlbru Cont d M g 1 Rotatonal axs x n x/2 M B g D M g Deterne the poston of the chld to balance the syste for dfferent poston of axs of rotaton. The net torque about the axs of rotaton by all the forces are τ M B g x / 2 + M g x / 2 n x / 2 M D g x / 2 0 Snce the noral force s The net torque can be rewrtten Therefore x τ n ( ) M B g + M g + M D g g x / 2 + M g ( x / 2) M B ( M g + M g + M g ) x / 2 M D g x / 2 B D M g M g x 0 M g M D g 350 PHYS , all 2008 Dr. Jaehoon D D What do we learn? No atter where the rotaton axs s, net effect of the torque s dentcal. 8

9 Ex for Mechancal Equlbru A person holds a 50.0N 0N sphere n hs hand. The forear s horzontal. The bceps uscle s attached 3.00 c fro the jont, and the sphere s 35.0c fro the jont. nd the upward force exerted by the bceps on the forear and the downward force exerted by the upper ar on the forear and actng at the jont. Neglect the weght of forear. O d U B l g ro the rotatonal equlbru condton Thus, the force exerted by the bceps uscle s orce exerted by the upper ar s Snce the syste s n equlbru, fro the translatonal equlbru condton x y 0 B U g 0 τ U 0 + B d g l 0 d g l B g l B 583N d 3.00 U B g N PHYS , all 2008 Dr. Jaehoon 9

10 Exaple 12 6 A 5.0 long ladder leans aganst a wall at a pont 4.0 above the ground. The ladder s unfor and has ass 12.0kg. Assung the wall s frctonless (but ground s not), deterne the forces exerted on the ladder by the ground and the wall. BD Gy O Gx W Thus, the y coponent of the force by the ground s Gy g rst the translatonal l equlbru, usng coponents x Gx W 0 y g + Gy 0 g N 118 The length x 0 s, fro Pythagoran theore x N PHYS , all 2008 Dr. Jaehoon 10

11 Exaple 12 6 cont d τ O ro the rotatonal equlbru g x W 0 Thus the force exerted on the ladder by the wall s g x W 44N The x coponent of the force by the ground s x Gx W 0 Solve for Gx Thus the force exerted on the ladder by the ground s 2 2 G Gx + Gy 44N N Gy tan tan 70 Gx 44 The angle between the θ 1 ground force to the floor Gx W PHYS , all 2008 Dr. Jaehoon 11

12 Elastc Propertes of Solds We have been assung that t the objects do not change ther shapes when external forces are exertng on t. It ths realstc? No. In realty, the objects get defored as external forces act on t, though the nternal forces resst the deforaton as t takes place. Deforaton of solds can be understood n ters of Stress and Stran Stress: A quantty proportonal to the force causng the deforaton. Stran: Measure of the degree of deforaton It s eprcally known that for sall stresses, stran s proportonal to stress The constants of proportonalty are called Elastc Modulus Elastc Modulus stress stran Three types of Elastc Modulus 1. Young s odulus: Measure of the elastcty n a length 2. Shear odulus: Measure of the elastcty n an area 3. Bulk odulus: Measure of the elastcty n a volue PHYS , all 2008 Dr. Jaehoon 12

13 Young s Modulus Let s consder a long bar wth cross sectonal area A and ntal t l length L. L L f L + L ex After the stretch ex Tensle stress A:cross sectonal area TensleStress Young s Modulus s defned as What s the unt of Young s Modulus? Experental Observatons Y ex A Tensle stran Tensle Stress Tensle Stran ex n Tensle Stran ex L orce per unt area 1. or a fxed external force, the change n length s proportonal to the orgnal length 2. The necessary force to produce the gven stran s proportonal to the cross sectonal area Elastc lt: Maxu stress that can be appled to the substance before t becoes peranently defored PHYS , all 2008 Dr. Jaehoon A L L L Used to characterze a rod or wre stressed under tenson or copresson 13

14 Bulk Modulus Blk Bulk Mdl Modulus characterzes the response of a substance bt to unfor squeezng or reducton of pressure. After the pressure change V V Noral orce Surface Area the force apples Volue stress Pressure pressure If the pressure on an object changes by P /A, the object wll undergo a volue change V. Bulk Modulus s defned as Because the change of volue s reverse to change of pressure. B Volue Volue Stress Stran PHYS , all 2008 Dr. Jaehoon V A V A P V Copressblty s the recprocal of Bulk Modulus 14 V

15 Exaple for Sold s Elastc Property A sold brass sphere s ntally under noral atospherc pressure of 1.0x10 5 N/ 2. The sphere s lowered nto the ocean to a depth at whch the pressures s 2.0x10 7 N/ 2. The volue of the sphere n ar s By how uch ts volue change once the sphere s suberged? Snce bulk odulus s B P V V V PV The aount of volue change s V B ro table 12.1, bulk odulus of brass s 6.1x10 10 N/ 2 The pressure change P s P P f P Therefore the resultng volue change V s V V f V PHYS , all 2008 Dr. Jaehoon The volue has decreased. 15

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