PHYS Look over. examples 2, 3, 4, 6, 7, 8,9, 10 and 11. How To Make Physics Pay PHYS Look over. Examples: 1, 4, 5, 6, 7, 8, 9, 10,

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1 PHYS Look over Chapter 9 Sectos - Eamples:, 4, 5, 6, 7, 8, 9, 0, PHYS Look over Chapter 7 Sectos -8 8, 0 eamples, 3, 4, 6, 7, 8,9, 0 ad How To ake Phscs Pa We wll ow look at a wa of calculatg where the pool balls wll go. To do ths we wll also have to use the Law of Coservato of Eerg

2 The Ceter of ass So far we have bee treatg objects as partcles, havg mass but o sze. Ths s fe for trasatoal moto, where each pot o a object epereces the same dsplacemet. But eve whe a object rotates or vbrates as t moves, there s oe pot o the object, called the Ceter of ass, that moves the same wa that a sgle partcle subject to the same force would move. The Ceter of ass We ca balace the teeter-totter at the pot that we could replace all the mass wth just a partcle. Ths s the ceter of mass or sometmes called the ceter of gravt. C.. Ceter of ass -D We defe the ceter of mass for a sstem of two partcles as: m + C + m m + m m Where: C The posto of the Ceter of ass alog the - as. The posto of partcle # The posto of partcle # The mass of partcle # The mass of partcle #

3 Ceter of ass for a Partcles For partcles: C m + m + m m + m + m m m m The Ceter of ass 3-d I three dmesos we just eed to fd the locato of the Ceter of ass for all 3 coordates (,,z) separatel b: C m m z C C m z I vector otato the ceter of mass ca be wrtte as: r C r C ˆ + r ˆj + C r zc kˆ The Ceter of ass for a Rgd Object The umber of partcles (atoms) a rgd object s so large ad ther spacg so small that we ca treat such a object as though t had a cotuous dstrbuto of mass. Δm We ca subdvded the object to small elemets of mass Δm located appromatel at (,,z ) C Δm Δm Δm z C zc 3

4 Now let the elemets of mass be further subdvded so that the umber of elemets teds to ft: z Ceter of ass Equatos C lm Δ m 0 C C lm Δm 0 Δm Δm 0 lm Δm Δm z C z C C dm dm z dm oto of the Ceter of ass If ou roll a cue ball at a d bllard ball that s at rest, we epect that the two balls wll roll forward after mpact. You would be surprsed f both balls started rollg back at ou or f the balls rolled off at a rght agle to the orgal moto. We are use to the fact that the Ceter of ass of the two balls moves forward as f o collso had happeed at all. Newto s d Law for a Sstem of Partcles Although the ceter of mass s just a pot, t moves lke a partcle whose mass s equal to the total mass of the sstem ad we ca assg a posto, a veloct ad a accelerato to t. The equato of moto for the ceter of mass s: F F Et a C Where F s the vector sum of all the forces actg o the sstem. The forces eerted b oe part of the sstem o the others cacel each other out b Newto s 3 th Law. 4

5 Lear ometum of a Partcle The Lear ometum of a partcle s a measure of how hard t wll be to stop the partcle. It s defed as the product of the mass ad the veloct p m v The ut for Lear ometum s: kg m s ometum ad Newto s d Law I terms of Lear ometum Newto s secod law reads: The rate of chage of Lear ometum of a bod s proportoal to the resultg force actg o the bod ad s the drecto of that force. F lm 0 Δp dp dt Lear ometum of a Sstem of Partcles Each partcle wll have a veloct ad a Lear ometum : Istead of a sgle partcle f we have a sstem of partcles, wth masses (m, m,...m ). The partcles the sstem ma teract wth each other ad there ma be eteral forces actg o the sstem as well. p m v p m v, p, m v 5

6 p The total Lear ometum for the sstem wll be: Total p + p + + p m v + m v + m v Whch b our defto of the ceter of mass becomes: p Total v The total mometum of a sstem of partcles s equal to the product of the total mass of the sstem ad the veloct of the Ceter of mass. C Coservato of Lear ometum From Newto s d Law terms of Lear ometum we have for a sstem of partcles: F Et lm Δ t 0 Δp Total p dt d Total If there are o Eteral Forces the we get: lm 0 Δ p Total 0 or p Total a costat Coservato of Lear ometum p Total a costat Ths s the Prcple of the coservato of Lear ometum. The Lear ometum of the dvdual d partcles ma chage, but ther sum remas costat f there are o et eteral forces. The law of Coservato of Lear ometum holds true eve atomc ad uclear phscs, although Newtoa echacs does ot. Hece ths coservato law must be eve more fudametal the Newtoa echacs. 6

7 Coservato Prcples The Coservato of Lear ometum Prcple s the secod of the great coservato prcples that we have met so far, the frst beg the coservato of Eerg prcple. Coservato prcples are of theoretcal ad practcal mportace phscs because the are smple ad uversal Dfferet observers, each hs or her ow referece frame would all agree, f the watch the same chagg sstem, that the coservato laws appled to sstem. Eample )A ralroad car moves at a costat speed of 3.0 m/s uder a gra elevator. Gra drops to t at a rate of 540 kg/m. What force must be appled to the ralroad car, the absece of frcto, to keep t movg at a costat speed? Collsos We ca lear about objects of all kds b observg them as the collde wth each other. Objects of terest that we stud b watchg collsos rage from subatomc partcles whose masses are 0-7 kg to galaes, whose masses are o the order of 0 7 kg ad everda objects wth masses -betwee. 7

8 Before ad After The prcpal tools for aalzg collsos are the laws of Coservato of Eerg ad ometum. I a collso a relatvel large force acts o each of the colldg partcles for a relatvel short tme. The basc dea of a collso s that the moto of the colldg partcles ca be separated tme to before the collso ad after the collso Impulse F mp Assume that durg a collso F mp acts from t utl t f the: Δp Fave Δp F I ave mp t F ave t f Where I mp s the Impulse whch h s a vector that pots the drecto of the vector chage mometum ad s equal to the area uder the force tme curve. The Impulse s a measure of the stregth ad durato of the collso force. Impulse If the collso force s large compared to the eteral forces ad the collso force acts for a short tme the the mometum for the sstem of partcles s coserved. I ths case we ca the sa that the mometum of a sstem of partcles just before the partcles collde s equal to the mometum of the sstem just after the partcles. 8

9 Two Kds of Collsos Collsos are usuall classfed accordg to whether of ot Ketc Eerg s coserved the collso. ) Elastc Collsos- Ketc Eerg s coserved. I these collsos the objects do ot Stck Together at all. ) Ielastc Collso- Ketc Eerg s ot coserved. I these collsos the objects do Stck Together some what. A completel elastc collso s whe two objects completel stck together. All collso betwee gross objects are elastc to some etet. Collsos -Dmeso Before After m v m m v f m v f v Elastc Collsos -D p m v +m v m v f +m v f KE (/)m (v ) +(/)m (v ) (/)m (v f ) +(/)m (v f ) p Completel Ielastc Collsos -D m v +m v (m +m )v f Eample )Two ttaum spheres approach each other head-o wth the same speed ad collde elastcall. After the collso, oe of the spheres, whose mass s 300 g, remas at rest. What s the mass of the other sphere. 9

10 Eample 3 3)eteor Crater Arzoa s thought to have bee formed b the mpact of a meteor wth the Earth some 0, rs ago. The mass of the meteor s estmated at kg, ad ts speed at 700 m/s. What speed would such a meteor mpart to the Earth a head-o collso? Collsos -Dmesos m m v v θ f θ v Elastc Collsos -D f p m v m v f cos θ +m v f cos θ p 0m v f s θ +m v f s θ KE (/)m (v ) (/)m (v f ) +(/)m (v f ) p p Completel Ielastc Collsos -D m v (m +m )vcos θ 0(m +m )vs θ Eample 4 4)Two cars collde at a tersecto. Car has a mass of 00 kg ad s movg at a veloct of 95.0 km/hr due east ad car has a mass of 400 km/hr due orth. The cars stck together ad move off as oe at agle θ. a)what s the agle θ? b)what s the fal veloct of the combed cars. 0

11 Summar of Chapter 9 ometum of a object: Newto s secod law: Total mometum of a solated sstem of objects s coserved. Durg a collso, the colldg objects ca be cosdered to be a solated sstem eve f eteral forces est, as log as the are ot too large. ometum wll therefore be coserved durg collsos. Summar of Chapter 9, cot. I a elastc collso, total ketc eerg s also coserved. I a elastc collso, some ketc eerg s lost. I a completel elastc collso, the two objects stck together after the collso. The ceter of mass of a sstem s the pot at whch eteral forces ca be cosdered to act.