System of Particles: Center-of-Mass

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1 System of Paticles: Cente-of-ass The cente-of-mass of a system of paticles is the point that moes as though () all of the systems mass wee concentated thee and () all etenal foces wee applied thee. Location of the Cente-of-ass: mi m m m m i (al mass of the system) Cente-of-ass mi i m m m m i m m m m mi i i Velocity of the Cente-of-ass: (al momentum of the system) P pi m m m m i P m m m m pi i cceleation of the Cente-of-ass: (net foce acting on the system of paticles) Fnet a miai ma i. Field // Uniesity of Floida m a m a m a F net dp dt PHY 5 Page a

2 Cente-of-ass: Supeposition Supeposition: If object has mass and its cente-of-mass is located at and object has mass and its cente-of-mass is located at then the cente of mass of the system is located at Eample:. Field // Uniesity of Floida What is the -coodinate of the cente-of-mass of the cicula disk of adius with a cicula hole of adius as shown in the figue? ssume the disk has a unifom mass density ρ. nswe: / Let object be a unifom disk of adius, height h, and cente at y. Let object be the disk with the hole in it. Let object be a unifom disk with adius, height h, and cente at y, -. ( ) π hρ ( π () π ( ) ) hρ Unifom density ρ PHY 5 Page hole Disk Disk Disk

3 Cente-of-ass: Eample Fall Eam Poblem : The figue shows a cubical bo with each side consisting of a unifom metal plate of negligible thickness. Each of the fou sides hae mass,, and the has mass. The bo is open at the top (at L) and has edge length L. If the the -coodinate of the cente-of-mass is at C L/, what is? nswe: % ight: 4% ( 4 ) L 4 ( L / ) 4 L L 4 4 ( L / ) L 4 y. Field // Uniesity of Floida PHY 5 Page

4 Collisions in Dimension: Elastic Elastic Collision: n elastic collision is one in which the kinetic enegy is conseed (i.e. the initial al kinetic enegy is equal to the al kinetic enegy). If a pojectile with mass and speed taeling to the ight along the -ais collides with a taget paticle at est with mass, what ae the elocities of the two paticles afte they undego an elastic collision? KE p (momentum conseation) initial p ( ) () (enegy conseation) ) initial KE ( ) ( ( ( ) ) ) ( )( ) ( Diide eq. by eq. and multiply by () () Lab Fame efoe -ais Lab Fame fte -ais ote: a -b (a-b)(ab) (4) ultiply eq. by and add it to eq. (5) Inset eq. 5 into eq.4. Field // Uniesity of Floida PHY 5 Page 4

5 d Collisions: Completely Inelastic Completely Inelastic Collision (Lab Fame): pojectile with mass and speed taeling to the ight along the -ais collides with a taget paticle at est with mass. If the two paticles stick togethe to fom a single paticle of mass, what is its elocity? What is the elocity of the cente-of-mass of the two paticles system befoe the collision? What is the change in the kinetic enegy befoe and afte the collision? (momentum conseation) pinitial p ( ) V V V ΔKE KE f KEi. Field // Uniesity of Floida Completely Inelastic Collision (C Fame): C C C C P p p p p C C C P p p p ( ) V f In the C fame the initial two paticles hae equal and opposite momentum. They stick togethe and ae at est with eo momentum and eo kinetic enegy. This coesponds to the maimal loss of kinetic enegy consistent with momentum conseation. Lab Fame efoe Lab Fame fte C Fame efoe PHY 5 Page 5 V C Fame fte V -ais -ais -ais -ais

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