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# Chapter 9 Notes Physics 125

UA

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This 4 page Class Notes was uploaded by Nora Salmon on Friday February 20, 2015. The Class Notes belongs to Physics 125 at University of Alabama - Tuscaloosa taught by Prof. Andreas Piepke in Fall2015. Since its upload, it has received 173 views. For similar materials see Physics 1 w/Calculus in Physics 2 at University of Alabama - Tuscaloosa.

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Date Created: 02/20/15

Chapter 9 Center of Mass and Linear Momentum 91 What is Physics 0 This chapter covers how the complicated motion of a system of objects can be simplified if we determine the system s center of mass 92 The Center of Mass 0 A system s center of mass com is the point whose movement is defined with two important points I The point moves as though all the system s mass is concentrated there I The point moves as though all external forces acting on the system are applied there 0 COM of a twoparticle system I Consider a system of two particles with masses m1 and m2 separated by a distance d I The origin of an x axis that defines the distance coincides with the particle with mass m1 m Then xcom 2 m1m2 If the masses are equal then the com is halfway between the two particles Also the com must lie somewhere in the distance between the two particles it cannot be somewhere outside of the system I Consider a system with two particles of masses m1 and m2 If the coordinate system is shifted so m1 is not at x0 then what m x m x 0 New equatlon xcom m1m2 0 Systems of more than two particles I We can rewrite m1 mg as M where M is the total mass of the system I Now we can expand the com equation to include n particles m1x1m2x2mnxn M New equation xcom 0 Threedimensional com equation I If the particles are distributed in three dimensions the center of mass must be identified by three coordinates instead of just one I This gives an xcom ycom zcom I The three coordinates form the components of a vector equation that can be used to find the vectorial quantity of the center of mass 1 Vector equatlon rcom E 731 mm M is the total mass of the system 0 Solid bodies I We treat solid bodies as a continuous distribution of matter Chapter 9 Center of Mass and Linear Momentum The particles that comprise the body become differential mass elements dm the sums of the components Xcom ycom and Zcom become integrals and we have a new equation for the com 0 Xcomz fxdm O Ycomz fydm O Zcomz dem Where M is the mass of the solid object I Since we treat the bodies as having uniform composition these bodies then have uniform density mass per unit volume 93 Newton s Second Law for a System of Particles o F ma actually applies to the system s com not the system in general I Therefore Fnet Macom I In this way we can study the acceleration of a system by studying the acceleration of its com I We assume the mass M of the system is constant making the system a closed system I We assume Fnet is the sum only of external forces not internal ones This is super important 0 Consider two pool balls hitting each other the collision causes a change in internal forces but no external forces therefore Fnet remains the same so acom remains the same and the center of mass continues accelerating in the same direction as it did before the collision 94 Linear Momentum o The linear momentum of a particle is a vector quantity p that is defined as I 15 2 m1 where m is mass and V is the particle s velocity I The time rate of change of the momentum of a particle is equal to the net force acting on the particle and is in the direction of that force gt d1 Fnet 5 In other words the net external force on a particle changes the particle s linear momentum 95 Linear Momentum of a System of Particles o A system of particles has a total linear momentum which is the vector sum of the individual particles linear momenta a I The 11near momentum of a system 1s P M1360quot Chapter 9 Center of Mass and Linear Momentum o This equation means that the linear momentum of a system of particles is equal to the product of the total mass M of the system and the velocity of the center of mass 96 Collision and Impulse o The momentum of a particlelike body cannot change unless a net force changes it I We can push on a body to change its motion I We can collide one body with another body 0 Single collision I Example baseball and bat The collision is brief The ball experiences a force that completely stops its motion then reverses it The force that the ball experiences varies during the collision and changes the ball s linear momentum I The change in an object s momentum is called the impulse I of the collision f If F39Ctdt This can also be written in component form with x and y components of impulse separated I Impulse and Newton s 3rd Law Newton s third law tells us that the force on the bat has the same magnitude but opposite direction as the force on the ball Therefore the impulse on the bat has the same magnitude but the opposite direction as the impulse on the ball 97 Conservation of Linear Momentum 0 Suppose the net force Fnet acting on a system O isolated system and the system is closed I Then P the linear momentum is constant I This result is called the law of conservation of linear momentum I Remember momentum can be conserved when energy is not I In a 2 or 3D coordinate system the linear momentums of each separate component may change or stay the same independent of the other components Throwing a ball across a room there is a net external vertical force but no net external horizontal force so the horizontal component of P can t change but the vertical component does 98 Momentum and Kinetic Energy in Collisions o In closed systems we can determine the results of a collision without knowing the details of the collision like how much damage is done Chapter 9 Center of Mass and Linear Momentum 0 There are three main types of collisions I An elastic collision is one in which the total kinetic energy of the system is conserved the same before and after the collision I An inelastic collision is one where some energy is transferred from kinetic energy to other forms of energy like heat or sound energy Total KE is NOT conserved I If the bodies stick together after collision the collision is called completely inelastic

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