PHYS 443 Secton 003 Lectue # Wednesday, Nov. 7, 00 D. Jaehoon Yu. Gavtatonal eld. negy n Planetay and Satellte Motons 3. scape Speed 4. lud and Pessue 5. Vaaton of Pessue and Depth 6. Absolute and Relatve Pessue Today s hoewok s hoewok # due 6:00p, day, Dec. 6!! Wednesday, Nov. 7, 00 PHYS 443-003, all 00 D. Jaehoon Yu
Announceents Reebe the Te xa on Monday, Dec. 9 n the class Coves all ateal n chaptes 5 Revew on Wednesday, Dec. 4 Happy Thanksgvng!!! Wednesday, Nov. 7, 00 PHYS 443-003, all 00 D. Jaehoon Yu
The gavtatonal foce s a feld foce. The Gavtatonal eld The foce exsts evey pont n the space. If one wee to place a test object of ass at any pont n the space n the exstence of anothe object of ass M, the test object wll fll the gavtatonal foce, g g, exeted by M. Theefoe the gavtatonal feld g s defned as In othe wods, the gavtatonal feld at a pont n space s the gavtatonal foce expeenced by a test patcle placed at the pont dvded by the ass of the test patcle. So how does the ath s gavtatonal feld look lke? g GM g ˆ R g g Whee ˆ s the unt vecto pontng outwad fo the cente of the ath a away fo the ath s suface Close to the ath s suface Wednesday, Nov. 7, 00 PHYS 443-003, all 00 D. Jaehoon Yu 3
The Gavtatonal Potental negy What s the potental enegy of an object at the heght y fo the suface of the ath? Do you thnk ths would wok n geneal cases? Why not? U gy No, t would not. Because ths foula s only vald fo the case whee the gavtatonal foce s constant, nea the suface of the ath and the genealzed gavtatonal foce s nvesely popotonal to the squae of the dstance. OK. Then how would we genealze the potental enegy n the gavtatonal feld? Because gavtatonal foce s a cental foce, and a cental foce s a consevatve foce, the wok done by g the gavtatonal foce s ndependent of the path. f The path can be looked at as consstng of g R any tangental and adal otons. Tangental otons do not contbute to wok!!! Wednesday, Nov. 7, 00 PHYS 443-003, all 00 D. Jaehoon Yu 4
Moe on The Gavtatonal Potental negy Snce the gavtatonal foce s a adal foce, t only pefoed wok whle the path was adal decton only. Theefoe, the wok pefoed bythe gavtatonal foce that depends on the poston becoes dw Potental enegy s the negatve change of wok n the path U GM Snce the ath s gavtatonal foce s ( ) So the potental enegy functon becoes nfntely apat. Wednesday, Nov. 7, 00 PHYS 443-003, all 00 D. Jaehoon Yu U f U Snce potental enegy only attes fo dffeences, by takng the nfnte dstance as the ntal pont of the potental enegy, we get o any two patcles? whole path d ( )d G U o the W f ( ) The enegy needed to take the patcles f GM f U o any patcles? GM f GM U U, j d f U ( ) d d U, j 5
xaple 4.6 A patcle of ass s dsplaced though a sall vetcal dstance y nea the ath s suface. Show that n ths stuaton the geneal expesson fothe change n gavtatonal potental enegy s educed to the Ug y. Takng the geneal expesson of gavtatonal potental enegy The above foula becoes U U Snce the stuaton s close to R and f the suface of the ath y Theefoe, U becoes U GM R Snce on the suface of the ath the gavtatonal feld s GM g GM R ( ) f f GM R f The potental enegy becoes y GM f U g y Wednesday, Nov. 7, 00 PHYS 443-003, all 00 D. Jaehoon Yu 6
M negy n Planetay and Satellte Motons v Wednesday, Nov. 7, 00 PHYS 443-003, all 00 D. Jaehoon Yu Consde an object of ass ovng at a speed v nea a assve object of ass M (M>>). What s the total enegy? Systes lke the Sun and the ath o the ath and the Moon whose otons ae contaned wthn a closed obt s called Bound Systes. o a syste to be bound, the total enegy ust be negatve. Assung a ccula obt, n ode fo the object to be kept n the obt the gavtatonal foce ust povde the adal acceleaton. Theefoe fo Newton s second law of oton v The knetc enegy fo ths syste s Theefoe the total echancal enegy of the syste s K + U K + U GM GM GM v a GM v Snce the gavtatonal foce s consevatve, the total echancal enegy of the syste s conseved. 7
xaple 4.7 The space shuttle eleases a 470kg councaton satellte whle n an obt that s 80k above the suface of the ath. A ocket engne on the satellte boosts t nto a geosynchonous obt, whch s an obt n whch the satellte stays dectly ove a sngle locaton on the ath, How uch enegy dd the engne have to povde? 4 What s the adus of the geosynchonous obt? T day 8.64 0 s o Keple s 3 d π law T K 3 Whee K s K 4 GS 9.89 0 s / 4 3 GM Theefoe the geosynchonous adus s GS T 4 4 3 ( 8.64 0 ) ( 8.64 0 ) 3 3 4.3 0 7 K 4 4 9.89 0 9.89 0 Because the ntal poston befoe the boost s 80k The total enegy needed to boost the satellte at the geosynchonous adus s the dffeence of the total enegy befoe and afte the boost Wednesday, Nov. 7, 00 PHYS 443-003, all 00 D. Jaehoon Yu 5 R +.80 0 6.65 0 6 6.67 0 GM s GS 5.98 0 4 470 4.3 0 7 6.65 0 6.9 0 8 0 J
v f 0 at h ax scape Speed R h v M Consde an object of ass s pojected vetcally fo the suface of the ath wth an ntal speed v and eventually coes to stop v f 0 at the dstance ax. Because the total enegy s conseved Solvng the above equaton fo v, one obtans Theefoe f the ntal speed v s known, one can use ths foula to copute the fnal heght h of the object. In ode fo the object to escape ath s gavtatonal feld copletely, the ntal speed needs to be v esc Ths s called the escape speed. Ths foula s vald fo any planet o lage ass objects. K + U v Wednesday, Nov. 7, 00 PHYS 443-003, all 00 D. Jaehoon Yu GM R. 0 4 h v GM GM R R How does ths depend on the ass of the escapng object? ax GM ax ax R GM v R v R 6.67 0 5.98 0 6 6.37 0 / s.k / s Independent of the ass of the escapng object 9 4
What ae the thee states of atte? How do you dstngush the? What s a flud? lud and Pessue Wednesday, Nov. 7, 00 PHYS 443-003, all 00 D. Jaehoon Yu Sold, Lqud, and Gas By the te t takes fo a patcula substance to change ts shape n eacton to extenal foces. A collecton of olecules that ae andoly aanged and loosely bound by foces between the o by the extenal contane. We wll fst lean about echancs of flud at est, flud statcs. In what way do you thnk flud exets stess on the object subeged n t? lud cannot exet sheang o tensle stess. Thus, the only foce the flud exets on an object esed n t s the foces pependcula to the sufaces of the object. Ths foce by the flud on an object usually s expessed n thefo of P the foce on a unt aea at the gven depth, the pessue, defned as A xpesson of pessue fo an d nfntesal aea da by the foce d s P da What s the unt and Unt:N/ Specal SI unt fo denson of pessue? Pa N / D.: [M][L - ][T - ] pessue s Pascal Note that pessue s a scala quantty because t s the agntude of the foce on a suface aea A. 0
xaple 5. The attess of a wate bed s.00 long by.00 wde and 30.0c deep. a) nd the weght of the wate n the attess. The volue densty of wate at the noal condton (0 o C and at) s 000kg/ 3. So the total ass of the wate n the attess s ρ V 3 W M 000.00.00 0.300.0 0 kg Theefoe the weght of the wate n the attess s W g.0 0 3 9.8.8 0 4 N b) nd the pessue exeted by the wate on the floo when the bed ests n ts noal poston, assung the ente lowe suface of the attess akes contact wth the floo. Snce the suface aea of the attess s 4.00, the pessue exeted on the floo s P A g A.8 0 4.00 4 3.95 0 Wednesday, Nov. 7, 00 PHYS 443-003, all 00 D. Jaehoon Yu
Vaaton of Pessue and Depth Wate pessue nceases as a functon of depth, and the a pessue deceases as a functon of alttude. Why? h P 0 A Mg PA Snce the syste s n ts equlbu Theefoe, we obtan P Atosphec pessue P 0 s.00at.03 0 It sees that the pessue has a lot to do wth the total ass of the flud above the object that puts weght on the object. Let s consde a lqud contaned n a cylnde wth heght h and coss sectonal aea A esed n a flud of densty ρ at est, as shown n the fgue, and the syste s n ts equlbu. If the lqud n the cylnde s the sae substance as the flud, the ass of the lqud n the cylnde s M ρv ρah P ρgh + 0 5 Pa PA P 0 A Mg The pessue at the depth h below the suface of a flud open to the atosphee s geate than atosphec pessue by ρgh. What else can you lean fo ths? PA P0 A ρahg 0 Wednesday, Nov. 7, 00 PHYS 443-003, all 00 D. Jaehoon Yu
Pascal s Law and Hydaulcs A change n the pessue appled to a flud s tanstted undnshed to evey pont of the flud and to the walls of the contane. P P 0 + ρgh What happens f P 0 s changed? d The esultant pessue P at any gven depth h nceases as uch as the change n P 0. Ths s the pncple behnd hydaulc pessue. How? A A Theefoe, the esultant foce s Ths sees to volate soe knd of consevaton law, doesn t t? Snce the pessue change caused by the the foce appled on to the aea A s tanstted to the on an aea A. Wednesday, Nov. 7, 00 PHYS 443-003, all 00 D. Jaehoon Yu d A In othe wods, the foce get ultpled by A the ato of the aeas A /A s tanstted to the on an aea. No, the actual dsplaced volue of the d flud s the sae. And the wok done d by the foces ae stll the sae. P A 3 A
xaple 5. In a ca lft used n a sevce staton, copessed a exets a foce on a sall pston that has a ccula coss secton and a adus of 5.00c. Ths pessue s tanstted by a lqud to a pston that has a adus of 5.0c. What foceust the copessed a exet to lft a ca weghng 3,300N? What a pessue poduces ths foce? Usng the Pascal s law, one can deduce the elatonshp between the foces, the foce exeted by the copessed a s A π ( 0.5 ) A π ( 0.05 ) Theefoe the necessay pessue of the copessed a s.33 0 4.48 0 3 N P A.48 0 π ( 0.05 ) 3.88 0 5 Pa Wednesday, Nov. 7, 00 PHYS 443-003, all 00 D. Jaehoon Yu 4
xaple 5.3 state the foce exeted on you eadu due to the wate above when you ae swng at the botto of the pool wth a depth 5.0. We fst need to fnd out the pessue dffeence that s beng exeted on the eadu. Then estate the aea of the eadu to fnd out the foce exeted on the eadu. Snce the outwad pessue n the ddle of the eadu s the sae as noal a pessue P P 0 ρ W gh 4 000 9.8 5.0 4.9 0 Pa statng the suface aea of the eadu at.0c.0x0-4, we obtan ( P )A 4 4 4.9 0.0 0 4.9N P 0 Wednesday, Nov. 7, 00 PHYS 443-003, all 00 D. Jaehoon Yu 5
H h dy y xaple 5.4 Wate s flled to a heght H behnd a da of wdth w. Detene the esultant foce exeted by the wate on the da. Snce the wate pessue vaes as a functon of depth, we wll have to do soe calculus to fgue out the total foce. The pessue at the depth h s Theefoe the total foce exeted by the wate on the da s P The nfntesal foce d exetng on a sall stp of da dy s d ρgh ρg( H y) PdA ρg( H y)wdy y H y 0 ρg ( H y) wdy ρg Hy y y H y 0 ρgh Wednesday, Nov. 7, 00 PHYS 443-003, all 00 D. Jaehoon Yu 6