PHYS 1443 Section 002


 Gwendolyn Holmes
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1 PHYS 443 Secton 00 Lecture #6 Wednesday, Nov. 5, 008 Dr. Jae Yu Collsons Elastc and Inelastc Collsons Two Dmensonal Collsons Center o ass Fundamentals o Rotatonal otons Wednesday, Nov. 5, 008 PHYS PHYS 4400, , Summer Fall 008 Dr. Dr. Yu
2 Announcements Quz net onday, Nov. 0 Begnnng o the class Covers CH 9 dterm grade dscussons I you haven t done t, please do so today ater the class n my oce, CPB34 Thrd term eam :0pm, Wednesday, Nov. 9, n SH03 Covers CH 9 What we complete net Wednesday, Nov. Jason wll do a summary sesson on onday, Nov. 7 Tea tme wth Dr. Durrance, a ormer astronaut, n SH08 at 3pm ths aternoon Wednesday, Nov. 5, 008 PHYS 4400, Summer 008 Dr.
3 EtraCredt Specal Project Derve the ormula or the nal velocty o two objects whch underwent an elastc collson as a uncton o known quanttes m, m, v 0 and v 0 n page 6 o ths lecture note. ust be done n ar greater detal than what s covered n the lecture note. 0 ponts etra credt Descrbe n detal what happens to the nal veloctes m m. 5 pont etra credt Due: Start o the class net Wednesday, Nov. Wednesday, Nov. 5, 008 PHYS 4400, Summer 008 Dr. 3
4 Collsons Generalzed collsons must cover not only the physcal contact but also the collsons wthout physcal contact such as that o electromagnetc ones n a mcroscopc scale. Consder a case o a collson between a proton on a helum on. F F Assumng no eternal orces, the orce object eerted on object by, F t, changes the momentum o object by F Usng Newton s 3 rd law we obtan The collsons o these ons never nvolve physcal contact because the electromagnetc repulsve orce between these two become great as they yget closer causng a collson. dp dp F dt F dt Lkewse or object by object dp So the momentum change o the system n the collson s 0, and the momentum s conserved Wednesday, Nov. 5, 008 F dt dp F dt dp dp + dp p + p p system PHYS 4400, Summer 008 Dr. 0 constant 4
5 Elastc and Inelastc Collsons omentum s conserved n any collsons as long as eternal orces are neglgble. bl Collsons are classed as elastc or nelastc based on whether the knetc energy s conserved, meanng whether t s the same beore and ater the collson. Elastc Collson Inelastc Collson A collson n whch the total knetc energy and momentum are the same beore and ater the collson. A collson n whch the total knetc energy s not the same beore and ater the collson, but momentum s. Two types o nelastc collsons:perectly nelastc and nelastc Perectly Inelastc: Two objects stck together ater the collson, movng together at a certan velocty. Inelastc: Colldng objects do not stck together ater the collson but some knetc energy s lost. Note: omentum s constant t n all collsons but knetc energy s only n elastc collsons. Wednesday, Nov. 5, 008 PHYS 4400, Summer 008 Dr. 5
6 Elastc and Perectly Inelastc Collsons In perectly Inelastc collsons, the objects stck together ater the collson, movng together. omentum s conserved n ths collson, so the nal velocty o the stuck system s How about elastc collsons? mv + mv In an elastc collson, both the momentum and the knetc energy are conserved. Thereore, the nal speeds n an elastc collson can be obtaned n terms o ntal speeds as m m m v + v m m + m + m m v + m mv + mv ( m+ m) v mv + mv v ( m+ m) mv + mv v m v + m ( v v ) m ( v v ) ( v v )( v v ) ( v v )( v v ) m + From momentum conservaton above m m m + + m v m + ( v v ) m ( v v ) m v v v + v m m m m Wednesday, Nov. 5, 008 PHYS 4400, Summer 008 Dr. What happens when the two masses are the same? 6
7 Eample or Collsons A car o mass 800kg stopped at a trac lght s rearended ended by a 900kg car, and the two become entangled. I the lghter car was movng at 0.0m/s beore the collson what s the velocty o the entangled cars ater the collson? Beore collson m 0.0m/s m Ater collson v m The momenta beore and ater the collson are p p mv + mv mv + mv 0 +mv m + m v ( ) Snce momentum o the system must be conserved p p ( m+ m) v mv m What can we learn rom these equatons on the drecton and magntude o the velocty beore and ater the collson? Wednesday, Nov. 5, 008 v m + m m v m / s ( ) The cars are movng n the same drecton as the lghter car s orgnal drecton to conserve momentum. The magntude s nversely proportonal to ts own mass. PHYS 4400, Summer 008 Dr. 7
8 Two dmensonal Collsons In two dmenson, one needs to use components o momentum and apply momentum conservaton to solve physcal problems. mv + m v mv + m v v v m comp. mv + mv φ θ m ycomp. mv + m v y y mv + m v mv + m v y y Consder a system o two partcle collsons and scatters n two dmenson as shown n the pcture. (Ths s the case at ed target accelerator eperments.) The momentum conservaton tells us: mv + mv m v mv φ m v + m v m v cosθ + m v cosφ m v y 0 m v y + mv y mv snθ mv snφ And or the elastc collsons, the knetc energy s conserved: Wednesday, Nov. 5, 008 m v m v + m PHYS 4400, Summer 008 Dr. v What do you thnk we can learn rom these relatonshps? 8
9 Eample or Two Dmensonal Collsons Proton # wth a speed m/s colldes elastcally wth proton # ntally at rest. Ater the collson, proton # moves at an angle o 37 o to the horzontal as and proton # delects at an angle φ to the same as. Fnd the nal speeds o the two protons and the scatterng angle o proton #, φ. v Snce both the partcles are protons m m m p. m Usng momentum conservaton, one obtans φ θ m comp. ycomp. m pv m pv cos θ + m pv cos φ m pv sn θ m pv sn φ 0 Cancelng m p and puttng n all known quanttes, one obtans ο v cos37 + v cosφ From knetc energy conservaton: 5 v.80 0 m / s Solvng Eqs. 3 Do ths at v + v equatons, one gets v. 0 m / s home ( ) (3) Wednesday, Nov. 5, 008 ο v sn 37 v sn φ PHYS 4400, Summer 008 Dr. φ () 53.0 ο 5 () 9
10 Center o ass We ve been solvng physcal problems treatng objects as szeless ponts wth masses, but n realstc stuatons objects have shapes wth masses dstrbuted throughout the body. Center o mass o a system s the average poston o the system s mass and represents the moton o the system as all the mass s on the pont. What does above statement tell you concernng the orces beng eerted on the system? The total eternal orce eerted on the system o total mass causes the center o mass to move at an acceleraton gven by a F / as all the mass o the system s concentrated on the center o mass. Consder a massless rod wth two balls attached at ether end. m m The poston o the center o mass o ths system s the mass averaged poston o the system m + m s closer to the m + m heaver object Wednesday, Nov. 5, 008 PHYS 4400, Summer 008 Dr. 0
11 oton o a Dver and the Center o ass Dver perorms a smple dve. The moton o the center o mass ollows a parabola snce t s a projectle moton. Dver perorms a complcated dve. The moton o the center o mass stll ollows the same parabola snce t stll s a projectle moton. Wednesday, Nov. 5, 008 PHYS 4400, Summer 008 Dr. The moton o the center o mass o the dver s always the same.
12 Eample 9 4 Thee people o roughly equvalent mass on a lghtweght (arlled) banana boat st along the as at postons.0m, 5.0m, and 3 6.0m. Fnd the poston o Wednesday, Nov. 5, 008 PHYS 4400, Summer 008 Dr. Usng the ormula or m m 4.0( m)
13 y m (0,) Eample or Center o ass n D A system conssts o three partcles as shown n the gure. Fnd the poston o the center o mass o ths system. Usng the ormula or or each poston vector component m m y (0.75,4) r y m m (,0) (,0) m m 3 One obtans r ( m+ m3) + mj +y j m m + m+ m3 m + m + m33 m + m3 m m I m kg; m m3 kg + m + m3 m + m + m3 m y m m j y + m y + m3 y3 m y r j m m + m + m3 m + m + m3 4 Wednesday, Nov. 5, 008 PHYS 4400, Summer 008 Dr. 3
14 r Center o ass o a Rgd Object The ormula or can be etended d to a system o many partcles or a Rgd Object m m + m + + m n n m+ m+ + mn m The poston vector o the center o mass o a many partcle system s r m Wednesday, Nov. 5, 008 y y j z k m r r + + r A rgd body an object wth shape and sze wth mass spread throughout the body, ordnary objects can be consdered as a group o partcles wth lm m 0 mass m densely spread throughout the gven shape o the object PHYS 4400, Summer 008 Dr. m y m z z m m r m m + my j+ mzk m m rdm dm 4
15 Eample o Center o ass; Rgd Body Show that the center o mass o a rod o mass and length L les n mdway between ts ends, assumng the rod has a unorm mass per unt length. L The ormula or o a contnuous object s L 0 dm Snce the densty o the rod (λ) s constant; d dmλd The mass o a small segment λ / L dm λd Thereore L L L d λ 0 λ 0 λl L Fnd the when the densty o the rod nonunorm but vares lnearly as a uncton o, λα L L αd 0 L 0 α L λ d 0 λd L α d L 0 α 0 L 3 αl L 3 3 Wednesday, Nov. 5, 008 PHYS 4400, Summer 008 Dr. α 3 L L 0
16 Center o ass and Center o Gravty The center o mass o any symmetrc object les on the as o symmetry and on any plane o symmetry, the object s mass s evenly dstrbuted throughout the body. How do you thnk you can determne the o the objects that are not symmetrc? Center o Gravty m g m Wednesday, Nov. 5, 008 As o One can use gravty to locate. symmetry. Hang the object by one pont and draw a vertcal lne ollowng a plumbob.. Hang the object by another pont and do the same. 3. The pont where the two lnes meet s the. Snce a rgd object can be consdered as a collecton o small masses, one can see the total gravtatonal orce eerted on the object as F F g What does ths equaton tell you? F m g PHYS 4400, Summer 008 Dr. g The net eect o these small gravtatonal orces s equvalent to a sngle orce actng on a pont (Center o Gravty) wth mass. The CoG s the pont n an object as all the gravtatonal orce s actng on! 6
17 oton o a Group o Partcles W We ve learned that the o a system can represent the moton o a system. Thereore, or an solated system o many partcles n whch the total mass s preserved, the velocty, total momentum, acceleraton o the system are Velocty o the system Total omentum o the system Acceleraton o the system The eternal orce eertng on the system v p a F et dr dt d mr dr m dt dt v dv dt F I net eternal orce s 0 Fet 0 Wednesday, Nov. 5, 008 a mv mv mv p ptot d mv dv m dt ma dt dp dt ma tot dp p tot p const dt tot PHYS 4400, Summer 008 Dr. What about the nternal orces? System s s momentum s conserved. 7
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