Ch09 Lecture Centre Of Mass And Linear Momentum

Chapter 9Center of Mass andLinear Momentum

9 Center of Mass and Linear MomentumEnergy& Rigid BodiesJ (Ns)April 25, 2022since mass is conserved velocities are changedPHY101 Physics I Dr.Cem Özdoğan2

9.2 The Center of Mass: System of ParticlesThe motion of rotating objects can be complicated(imagine flipping a baseball bat into the air). But there is a special point on the object forwhich the motion is simple. The center of mass (black dot) of a baseball batflipped into the air follows a parabolic path, butall other points of the bat follow morecomplicated curved paths. The center of mass (COM) of a system ofparticles is the point that moves as though1.All of the system’s mass were concentrated there.2.All external forces were applied there. object a point like particleFext at COMFor two particles separated by adistance d, where the origin is chosenat the position of particle 1: x 0 &1x2 dApril 25, 2022PHY101 Physics I Dr.Cem Özdoğan3

9.2 The Center of Mass: System of Particles For two particles, for an arbitrary choice of origin: For many particles, we can generalize the equation. Considera situation in which n particles are strung out along the x axis.Let the mass of the particles are m1, m2, .mn, and let them belocated at x1, x2, xn respectively. Then if the total mass is M m m . m , then the location of the12ncenter of mass, xcom, is XCOM, YCOM, ZCOMApril 25, 2022PHY101 Physics I Dr.Cem Özdoğan4

9.2 The Center of Mass: System of ParticlesThe center of mass is in the same location regardless of the coordinatesystem used. It is a property of the particles, not the coordinates. In three dimensions, we find the center of mass along each axisseparately: The position of the center of mass can be expressed in vector notation as:April 25, 2022PHY101 Physics I Dr.Cem Özdoğan5

9.2 The Center of Mass: Solid Body For solid bodies, we take the limit of an infinite sum of infinitely smallparticles integration! (m Δm dm)where M is the mass of the object. If the object has uniform density, ρ,defined as: We limit ourselves to objects of uniform density, ρ, for the sake ofsimplicity, thenwhere V is the volume of the object. You can bypass one or more ofthese integrals if the object has symmetry.April 25, 2022PHY101 Physics I Dr.Cem Özdoğan6

9.2 The Center of Mass: System of ParticlesSample problem: COM of 3 particlesThe total mass M of the system is 7.1 kg.The coordinates of the center of mass aretherefore:We are given the following data:Note that the zcom 0.April 25, 2022PHY101 Physics I Dr.Cem Özdoğan7

9.3 Newton’s 2nd Law for a System of Particles Center of mass motion continues unaffected by forces internal to asystem (collisions between billiard balls).The vector equation that governs the motion of the center of mass of sucha system of particles is:Note that:1. Fnet is the net force of all externalforces that act on the system. Forceson one part of the system fromanother part of the system (internalforces) are not included.2. M is the total mass of the system. Mremains constant, and the system issaid to be closed.3. acom is the acceleration of the centerof mass of the system.April 25, 2022PHY101 Physics I Dr.Cem Özdoğan8

9.3 Newton’s 2nd Law for a System of Particles: Proof of Final Result For a system of n particles,where M is the total mass, and ri are the position vectors of the masses mi. Differentiating,where the v vectors are velocity vectors. This leads to Finally, FnetWhat remains on the right hand side is the vector sum of all the external forces that act onthe system, while the internal forces cancel out by Newton’s 3 rd Law.April 25, 2022PHY101 Physics I Dr.Cem Özdoğan9

9.3 Newton’s 2nd Law for a System of ParticlesSample problem: Motion of the COM of 3 ParticlesCalculations:ApplyingNewton’s secondlaw to the centerof mass,April 25, 2022PHY101 Physics I Dr.Cem Özdoğan10

9.4 Linear MomentumDEFINITION:linear: since not rotationin which m is the mass of the particle and v is its velocity. The time rate of change of the momentum of a particle is equal to thenet force acting on the particle and is in the direction of that force.Manipulating this equation:(Newton’s 2nd Law)April 25, 2022PHY101 Physics I Dr.Cem Özdoğan11

9.5 Linear Momentum of a System of Particles The linear momentum of a system of particles is equal to the productof the total mass M of the system and the velocity of the center ofmass.The time rate of change of the momentum of a particle is equal to thenet force acting on the particle and is in the direction of that force.The net external force on a system changes linear momentum. Without a net external force, the total linear momentum of a systemof particles cannot change. April 25, 2022PHY101 Physics I Dr.Cem Özdoğan12

9.6 Collision and Impulse In this case, the collision is brief, and the ballexperiences a force that is great enough toslow, stop, or even reverse its motion. The figure depicts the collision at one instant. The ball experiences a force F(t) that variesduring the collision and changes the linearmomentum of the ball.April 25, 2022PHY101 Physics I Dr.Cem Özdoğan13

9.6 Collision and Impulse In a collision, momentum of a particle can change. The change in linear momentum is related to the force by Newton’ssecond law written in the form The right side of the equation is a measure of both the magnitude andthe duration of the collision force, and is called the impulse of thecollision, J (Unit: Ns). This means that the applied impulse is equal to the change inmomentum of the object during the collision:April 25, 2022PHY101 Physics I Dr.Cem Özdoğan14

9.6 Collision and Impulse We are integrating:we only need toknow the areaunder the forcecurve! Instead of the ball, one can focus on the bat. At any instant, Newton’s third law says that theforce on the bat has the same magnitude but theopposite direction as the force on the ball. That means that the impulse on the bat has thesame magnitude but the opposite direction as theimpulse on the ball.April 25, 2022PHY101 Physics I Dr.Cem Özdoğan15

9.6 Collision and ImpulseSample problem: 2-D ImpulseApril 25, 2022PHY101 Physics I Dr.Cem Özdoğan16

9.6 Collision and ImpulseSample problem: 2-D Impulsecontd.April 25, 2022PHY101 Physics I Dr.Cem Özdoğan17

9.7 Conservation of Linear Momentum If no net external force acts on a system of particles, the total linearmomentum, P, of the system cannot change. (an impulse of zero )If the component of the net external force on a closed system is zeroalong an axis, then the component of the linear momentum of the systemalong that axis cannot change.This is called the law of conservation of linear momentum. Internal forces can change momenta of parts of the system, but cannotchange the linear momentum of the entire system. April 25, 2022PHY101 Physics I Dr.Cem Özdoğan18

9.7 Conservation of Linear MomentumSample problem: 2-D ExplosionApril 25, 2022PHY101 Physics I Dr.Cem Özdoğan20

9.8 Momentum and Kinetic Energy in Collisions1. In a closed and isolated system, if there are two colliding bodies, andtotal kinetic energy is unchanged (conserved) by the collision. Such a collision is called an elastic collision. A useful approximation for common situations. In real collisions, some energy is always transferred.2. If during the collision, some energy is always transferred from kineticenergy to other forms of energy, such as thermal energy or energy ofsound, then the kinetic energy of the system is not conserved. Such a collision is called an inelastic collision. Some energy is transferred.3. Completely inelastic collisions: The objects stick together Greatest loss of kinetic energy.April 25, 2022PHY101 Physics I Dr.Cem Özdoğan21

9.9 Inelastic collisions in 1-DInelastic collision:Completely inelastic collision, fortarget at rest: m2v2i 0 & v1f v2f Vin x-directionApril 25, 2022PHY101 Physics I Dr.Cem Özdoğan22

9.9 Inelastic collisions in 1-DThe center ofmass velocityremainsunchanged:Fig. 9-16 Some freeze framesof a two-body system, which undergoes acompletely inelastic collision. The system’scenter of mass is shown in each freeze-frame. April 25, 2022The velocity vcom of the center ofmass is unaffected by thecollision.Because the bodies stick togetherafter the collision, their commonvelocity V must be equal to vcom.PHY101 Physics I Dr.Cem Özdoğan23

9.9 Inelastic collisions in 1-DSample problem: conservation ofmomentumThe collision within the bullet– block system isso brief. Therefore:(1)During the collision, the gravitational force onthe block and the force on the block from thecords are still balanced. Thus, during thecollision, the net external impulse on the bulletblock system is zero. Therefore, the system isisolated and its total linear momentum isconserved.(2) The collision is one-dimensional in the sensethat the direction of the bullet and block just afterthe collision is in the bullet’s original direction ofmotion.As the bullet and block now swing uptogether, the mechanical energy of the bullet–block–Earth system is conserved:Combining steps:April 25, 2022PHY101 Physics I Dr.Cem Özdoğan24

9.10 Elastic collisions in 1-D: Stationary TargetIn an elastic collision, the kinetic energy of each colliding body maychange, but the total kinetic energy of the system does not change.With some algebra we get:Results:Equal masses, v1f 0, v2f v1i: the first object stops.Massive target, m2 m1: the first object just bounces back, speed mostly unchanged.Massive projectile, v1f v1i, v2f 2v1i: the first object keeps going, the target fliesforward at about twice its speed. oooApril 25, 2022PHY101 Physics I Dr.Cem Özdoğan25

9.10 Elastic collisions in 1-D: Moving TargetFor a target that is also moving,With some algebra we get:April 25, 2022PHY101 Physics I Dr.Cem Özdoğan26

9.10 Elastic collisions in 1-DSample problem: Two PendulumsApril 25, 2022PHY101 Physics I Dr.Cem Özdoğan27

9.11 Collisions in 2-DIf elastic,Apply the conservation of momentumalong each axis:x:y:Also, apply conservation of energy forelastic collisions:known(s) unknown(s) !April 25, 2022PHY101 Physics I Dr.Cem Özdoğan28

9 Solved Problems1. A shell is shot with an initial velocity of v0 20 m/s, at an angle of θ 60o with thehorizontal. At the top of the trajectory, the shell explodes into two fragments ofas max heightequal mass (see Figure). One fragment, whose speed immediately after theexplosion is zero, falls vertically. How far from the gun does the other fragmentland, assuming that the terrain is level and that air drag is negligible?April 25, 2022PHY101 Physics I Dr.Cem Özdoğan29

9 Solved ProblemsApril 25, 2022PHY101 Physics I Dr.Cem Özdoğan31

9 Solved ProblemsApril 25, 2022PHY101 Physics I Dr.Cem Özdoğan33

9 Solved ProblemsApril 25, 2022PHY101 Physics I Dr.Cem Özdoğan35

9 Solved ProblemsApril 25, 2022PHY101 Physics I Dr.Cem Özdoğan37

9 Solved ProblemsApril 25, 2022PHY101 Physics I Dr.Cem Özdoğan39

9 Solved ProblemsApril 25, 2022PHY101 Physics I Dr.Cem Özdoğan41

9 Solved ProblemsApril 25, 2022PHY101 Physics I Dr.Cem Özdoğan43

9 SummaryLinear Momentum & Newton's 2nd Motion of the Center of MassUnaffected by collisions/internal forcesLaw Linear momentum defined as:Collisions in Two DimensionsEq. (9-25) Write Newton's 2nd law:Apply conservation of momentumalong each axis individually Conserve K if elastic Eq. (9-27)Conservation of Linear Momentum Elastic Collisions in One DimensionEq. (9-42)K is also conserved Eq. (9-67)Collision and Impulse Defined as:Eq. (9-68)Eq. (9-30)Impulse causes changes inlinear momentumVariable-Mass SystemsInelastic Collision in 1D Momentum conserved along thatdimensionEq. (9-51)Eq. (9-87)Eq. (9-88)April 25, 2022PHY101 Physics I Dr.Cem Özdoğan44

9 Center of Mass and Linear MomentumEnergyAdditional MaterialsApril 25, 2022PHY101 Physics I Dr.Cem Özdoğan45