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UT Arlington PHYS 1443 - Linear Momentum

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PHYS 1443 – Section 001 Lecture #11AnnouncementsLinear MomentumLinear Momentum and ForcesConservation of Linear Momentum in a Two Particle SystemLinear Momentum ConservationMore on Conservation of Linear Momentum in a Two Particle SystemExample for Linear Momentum ConservationImpulse and Linear MomentumExample 9-6Example 9 – 6 cont’dAnother Example for ImpulseCollisionsElastic and Inelastic CollisionsElastic and Perfectly Inelastic CollisionsExample for CollisionsTwo dimensional CollisionsExample for Two Dimensional CollisionsTuesday, June 20, 2006 PHYS 1443-001, Summer 2006Dr. Jaehoon Yu1PHYS 1443 – Section 001Lecture #11Tuesday, June 20, 2006Dr. Jaehoon Yu•Linear Momentum•Linear Momentum and Forces•Conservation of Momentum•Impulse and Momentum Change•Collisions•Two Dimensional Collision s•Center of MassToday’s homework is HW #6, due 7pm, Friday, June 23!!Tuesday, June 20, 2006 PHYS 1443-001, Summer 2006Dr. Jaehoon Yu2Announcements•Quiz this Thursday–Eerly in the class–Covers Ch. 8.5 – Ch. 9•Mid-term grade discussions tomorrow–Bottom half of the classTuesday, June 20, 2006 PHYS 1443-001, Summer 2006Dr. Jaehoon Yu3Linear MomentumThe principle of energy conservation can be used to solve problems that are harder to solve just using Newton’s laws. It is used to describe motion of an object or a system of objects.A new concept of linear momentum can also be used to solve physical problems, especially the problems involving collisions of objects.p �rLinear momentum of an object whose mass is m and is moving at a velocity of v is defined as What can you tell from this definition about momentum?What else can use see from the definition? Do you see force?The change of momentum in a given time intervalptD=Dr0mv mvt-=Dr r( )0m v vt-=Dr rvmtD=DrF�rma =r1. Momentum is a vector quantity.2. The heavier the object the higher the momentum3. The higher the velocity the higher the momentum4. Its unit is kg.m/s mvrTuesday, June 20, 2006 PHYS 1443-001, Summer 2006Dr. Jaehoon Yu4Linear Momentum and ForcesWhat can we learn from this Force-momentum relationship?Something else we can do with this relationship. What do you think it is?F =�rThe relationship can be used to study the case where the mass changes as a function of time.Can you think of a few cases like this?Motion of a meteoriteMotion of a rocket •The rate of the change of particle’s momentum is the same as the net force exerted on it.•When net force is 0, the particle’s linear momentum is constant as a function of time.•If a particle is isolated, the particle experiences no net force. Therefore its momentum does not change and is conserved.dpFdt=�rr( )d mvdt=rdmvdt=rdvmdt+rdpdtrTuesday, June 20, 2006 PHYS 1443-001, Summer 2006Dr. Jaehoon Yu5Conservation of Linear Momentum in a Two Particle SystemConsider an isolated system with two particles that does not have any external forces exerting on it. What is the impact of Newton’s 3rd Law?Now how would the momenta of these particles look like?If particle#1 exerts force on particle #2, there must be another force that the particle #2 exerts on #1 as the reaction force. Both the forces are internal forces and the net force in the entire SYSTEM is still 0. Let say that the particle #1 has momentum p1 and #2 has p2 at some point of time.Using momentum-force relationship121 dpFdt=rrAnd since net force of this system is 02 1p p const+ =r rThereforeF�rThe total linear momentum of the system is conserved!!!and12 21F F= +r r2 1dp dpdt dt= +r r( )2 1dp pdt= +r r0212 dpFdt=rrTuesday, June 20, 2006 PHYS 1443-001, Summer 2006Dr. Jaehoon Yu6Linear Momentum Conservation1 2i ip p+ =r r1 2f fp p+ =r r1 1 2 2m v m v+r r1 1 2 2m v m v� �+r rTuesday, June 20, 2006 PHYS 1443-001, Summer 2006Dr. Jaehoon Yu7More on Conservation of Linear Momentum in a Two Particle SystemWhat does this mean?As in the case of energy conservation, this means that the total vector sum of all momenta in the system is the same before and after any interactionMathematically this statement can be written as Whenever two or more particles in an isolated system interact, the total momentum of the system remains constant.p =�rFrom the previous slide we’ve learned that the total momentum of the system is conserved if no external forces are exerted on the system.2 1i ip p+ =r rThis can be generalized into conservation of linear momentum in many particle systems.systemxfsystemxiPPsystemyfsystemyiPPsystemzfsystemziPP2 1f fp p+r r2 1p p+ =r rconstTuesday, June 20, 2006 PHYS 1443-001, Summer 2006Dr. Jaehoon Yu8Example for Linear Momentum ConservationEstimate an astronaut’s resulting velocity after he throws his book to a direction in the space to move to a direction.iprFrom momentum conservation, we can writevAvBAssuming the astronaut’s mass is 70kg, and the book’s mass is 1kg and using linear momentum conservationAv =rNow if the book gained a velocity of 20 m/s in +x-direction, the Astronaut’s velocity isAv =rA A B Bm v m v= +r rB BAm vm- =r170Bv-r( )12070i- =r( )0.3 /i m s-r0fp=rTuesday, June 20, 2006 PHYS 1443-001, Summer 2006Dr. Jaehoon Yu9Impulse and Linear Momentum By integrating the above equation in a time interval ti to tf, one can obtain impulse I.Effect of the force F acting on a particle over the time interval t=tf-ti is equal to the change of the momentum of the particle caused by that force. Impulse is the degree of which an external force changes momentum.The above statement is called the impulse-momentum theorem and is equivalent to Newton’s second law. dpFdt=rrNet force causes change of momentum  Newton’s second lawSo what do you think an impulse is?What are the dimension and unit of Impulse? What is the direction of an impulse vector? Defining a time-averaged force 1iiF F tt� DD�r rImpulse can be rewritten I F t� Dr rIf force is constant I F t� Dr rIt is generally assumed that the impulse force acts on a short time but much greater than any other forces present.dp Fdt=rrfittdp =�rf ip p- =r rpD =rfittFdt =�rIrTuesday, June 20, 2006 PHYS 1443-001, Summer 2006Dr. Jaehoon Yu10Example 9-6(a) Calculate the impulse experienced when a 70 kg person lands on firm ground after jumping from a height of 3.0 m. Then estimate the average force exerted on the person’s feet by the ground, if the landing is (b) stiff-legged and (c) with bent legs. In the former case, assume the body moves


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UT Arlington PHYS 1443 - Linear Momentum

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