Note for PH 101 at UA-General Physics I (7)
Note for PH 101 at UA-General Physics I (7)
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Date Created: 02/06/15
GIANCOLI Lecture PowerPoints Chapter 7 Physics Principles with Applications 6 edition Giancoli 1 Chapter 7 39Linear Momentum V Copyright 2005 Pearson Prentice Hall Inc Units of Chapter 7 Momentum and Its Relation to Force Conservation of Momentum Collisions and Impulse Conservation of Energy and Momentum in Collisions Elastic Collisions in One Dimension Units of Chapter 7 nelastic Collisions Colisions in Two or Three Dimensions Center of Mass CM CM for the Human Body Center of Mass and Translational Motion Momentum After having learnt our first conservation law for Energy we will now learn another one for momentum This gives a powerful new way to analyse motion particularly problems involving collisions 71 Momentum and Its Relation to Force Momentum is a vector symbolized by the symbol p and is defined as 71 The rate of change of momentum is equal to the net force 72 This can be shown usmg Newton s second law 72 Conservation of Momentum During a collision measurements show that the total momentum does not change mA VA mB VB e i Copyright 2005 Pearson Prentice Hall Inc 72 Conservation of Momentum More formally the law of conservation of momentum states The total momentum of an isolated system of objects remains constant VA240 ms VB 0 at rest r 3 7 K L I v r V V V V V 1 l39lMlMh HIMl 39lll JIM i x new b After collision Copyright 2005 Pearson Prentice Hall Inc 72 Conservation of Momentum Momentum conservation works for a rocket as long as we consider the rocket and its fuel to be one system and account for the mass loss of the rocket is gas procket Copyright 2005 Pearson Prentice Hall Inc 73 Collisions and Impulse Copyright 2005 P eeee on Prentice Hall Inc During a collision objects are deformed due to the large forces involved 5 Ari Since F a we can At write if A AF 75 The definition of impulse impulse 1 5 At 73 Collisions and Impulse Since the time of the collision is very short we need not worry about the exact time dependence of the force and can use the average force Force F 0 39 t l 1me Cnpyrighl 2005 Pearson Prentice Hall Inc Copyright 2005 Pearson Prentice Hall In 73 Collisions and Impulse The impulse tells us that we can get the same change in momentum with a large force acting for a short time or a small force acting for a longer time This is why you should bend your knees when you land why airbags work and why landing on a pillow hurts less than landing on concrete v0 Copyright 2005 Pearson Prentlce Hall Inc 74 Conservation of Energy and Momentum in Collisions a Approach Momentum Is conserved in all collisions W 39 b Collision Collisions in which kinetic energy is W W conserved as well are C Ifelastic called elastic collisions and those In which It IS VA E E V B not are called inelastic d If inelastic Copyright 2005 Pearson Prentice Hall Inc 75 Elastic Collisions in One Dimension m m i gt 3 VA VB VA VB b Copyright 2005 Pearson Prentice Hall Inc Here we have two objects colliding elastically We know the masses and the initial speeds Since both momentum and kinetic energy are conserved we can write two equations This allows us to solve for the two unknown final speeds 76 Inelastic Collisions With inelastic collisions some of the initial kinetic energy is lost to thermal or potential energy It may also be gained during explosions as there is the addition of chemical or nuclear energy A completely inelastic collision is one where the objects stick ifs w together afterwards so there is l 1th only one final velocity 1 b Copyright 2005 Pearson Prentice Hall Inc 78 Center of Mass In a the diver s motion is pure translation in b it is translation plus rotation There is one point that moves in the same path a particle would or i take if subjected Fl to the same force as the diver This t point is called the center of mass CM 78 Comet ot ooo The general mottoo of out oojjoot coon loo oooo olotoo ao tho sum of the translational mottoo of the CME tomato rotational vibrational or other totmo ot mot oo otooot the t tn Copyright 2005 Pearson Prentice Hall Inc 78 Center of Mass For two particles the center of mass lies closer to the one with the most mass mAxA meB mAxA n szB xCM mA mB M where M is the total mass R as Y Copyright 2005 Pearson Prentice Hall Inc 78 Center of Mass The center of gravity is the point where the gravitational force can be considered to act It is the same as the center of mass as long as the gravitational force does not vary among different parts of the object P1vot pomt Copyright 2005 Pearson Prenti lllllllll c 78 Center of Mass The center of gravity can be found experimentally by suspending an object from different points The CM need not be within the actual object a doughnut s CM is in the center of the hole Copyright 2005 Pearson Prentice Hall Inc 79 CM for the Human Body The x s in the small diagram mark the CM of the listed body segments TABLE 7 1 Center of Mass of Parts of Typical Human Body full height and mass 100 units Distance Above Floor Hinge Points 0 Center of Mass x Percent of Hinge Points Joints Height Above Floor Mass 912 Base of skull Head 935 69 812 Shoulder joint Trunk and neck 711 461 am 512quot g Upper arms 717 66 E Lower arms 553 42 521 Hip joint m 4539 ands 431 17 Upper legs thighs 425 215 285 Knee joint Lower legs 182 96 40 Ankle joint 3 eet 18 34 Body CM 580 1000 Copyright 2005 Pearson Prentice Hall Inc 79 CM for the Human Body 4 503 The location of the center of mass of the leg circled will depend on the position of theleg b Copyright 2005 Pearson Prentice Hall Inc 79 CM for the Human Body 5 I J High jumpers have developed a technique where their CM actually passes under the bar as they go over it This allows them to clear higher bars sA Copyright 2005 Pearon Prentice Hall Inc Summary of Chapter 7 Momentum of an object 3 mi Newton s second law A P 2F At Total momentum of an isolated system of objects is conserved During a collision the colliding objects can be considered to be an isolated system even if external forces exist as long as they are not too large Momentum will therefore be conserved during collisions Summary of Chapter 7 cont Impulse i5 M AF In an elastic collision total kinetic energy is also conserved In an inelastic collision some kinetic energy is lost In a completely inelastic collision the two objects stick together after the collision The center of mass of a system is the point at which external forces can be considered to act Backup 77 Collisions in Two or Three Dimensions Conservation of energy and momentum can also be used to analyze collisions in two or three dimensions but unless the situation is very simple the math quickly becomes unwieldy Here a moving object collides with an object initially at rest Knowing a 9 450 the masses and initial VA A gt Aj x veOCltleS Is not enough B 45 we need to know the angles as well in order to find the final velocities Copyright 2005 Pearson Prentice Hall Inc 77 Collisions in Two or Three Dimensions Problem solving 1 Choose the system If it is complex subsystems may be chosen where one or more conservation laws apply 2 Is there an external force If so is the collision time short enough that you can ignore it 3 Draw diagrams of the initial and final situations with momentum vectors labeled 4 Choose a coordinate system 77 Collisions in Two or Three Dimensions 5 Apply momentum conservation there will be one equation for each dimension 6 If the collision is elastic apply conservation of kinetic energy as well 7 Solve 8 Check units and magnitudes of result 710 Center of Mass and Translational Motion The total momentum of a system of particles is equal to the product of the total mass and the velocity of the center of mass The sum of all the forces acting on a system is equal to the total mass of the system multiplied by the acceleration of the center of mass MamI Fnet 73911 710 Center of Mass and Translational Motion This is particularly useful in the analysis of separations and explosions the center of mass which may not correspond to the position of any particle continues to move according to the net force IF d 44 d 7 Copyright 2005 Pearson Prentice Hall Inc
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