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UW-Madison PHYSICS 107 - Momentum conservation, equal masses

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1From LastTime…• Position, velocity, and acceleration– velocity = time rate of change of position– acceleration = time rate of change of velocity– Particularly useful concepts when• velocity is constant (undisturbed motion)• acceleration is constant (free falling object)• Momentum and conservation of momentum– Descartes characterized changes in motion withmomentum = mass x velocity.– Momentum measures the amount of ‘motion’ of an object.– Total momentum of a system is conserved:momentum is just transferred between bodies.HW#2: Griffith Chap 7: Q19, Q20, E10, E14March Chap 2: Exercise 10March Chap 3, Exercises 2, 4Momentum conservation: equalmassesBefore collision:After collision:m = 1 kgv = 1 m/sp = mv = 1 kg-m/sm =1 kgv = 0 m/sp=0m =1 kgv = 0 m/sp = 0 kg-m/sm = 1 kgv = 1 m/sp = 1 kg-m/sTotal momentum =1 kg-m/s+0 kg-m/s = 1 kg-m/sTotal momentum =0 kg-m/s+1 kg-m/s = 1 kg-m/s1 212Momentum transfer• Ball 2 increased velocity from 0 m/s to 1 m/s.• Ball 2 increased its momentum by 1 kg-m/s. Thismomentum came from ball 1, which reduced itsmomentum by 1 kg-m/s.• We normally say that 1 kg-m/s of momentum wastransferred from ball 1 to ball 2.• The total momentum of the system (ball 1 + ball 2) isthe same before and after the collision.Impulse• In Griffith’s book (chapter 7),he discusses this as an impulse.• An impulse is a short ‘disturbance’ exerted on anobject.• It is equal to the momentum change of the object.Are we done yet?• Where does Newton come in? What is left to do?• Like Galileo and Descartes, Newton has a law of inertia.• Newton’s first law:Every body perseveres in its state of rest, or of uniformmotion in a right line, unless it is compelled to changethat state by forces impressed upon it.• The ‘force’ is the ‘external disturbance’ of Galileo andDescartesNewtonian Forces• Newton made a definition of force thatdescribed how momentum was transferred.• He viewed it as a continuous process rather thanthe immediate transfer as Descartes andGalileo.• This makes a connection with our intuitiveunderstanding of ‘force’ as a push or a pull.2Newton’s second law• The change in motion is proportional to themotive force impressed; and is made in thedirection of the right line in which that force isimpressed.Change in momentum = ΔpApplied force = F Δp = F ΔtTime interval = Δt(Momentum change)=(Applied force)×(time interval)Momentum = (mass) × (velocity)Change in momentum = (mass) × (change in velocity)(change in momentum)=(Applied force)×(change in time) ! "change in momentumchange in time = applied force ! " mass #change in velocitychange in time = applied forceaccelerationForce = (mass) × (acceleration) ! acceleration = forcemassNewton’ssecond lawForce results in acceleration• A body will accelerate (change its velocity)when another body exerts a force on it.• This is also a change in momentum.• But what is a force?– Push– Pull– Jet thrustF1More than one force…• Total force determines acceleration• If F1 and F2 balance, acceleration is zero.F1F2Back to falling bodies• A free-falling body moves with constantacceleration.• Newton says that this means there is a constantforce on the falling body.• This is the gravitational force, and is directeddownward.• The acceleration of falling bodies is experimentallyindependent of massQuestionWhen the vectron hovers near the ceiling, the propeller forcecompared to hovering near the floor isA. Greater.B. Less.C. The same.Gravity exerts a force downward. When the vectronhovers, its velocity is constant, so the acceleration iszero. This means the net force is zero. The propellerforce balances the gravitational force3Types of forcesThe Four Forces1. Strong nuclear force2. Electromagnetic force3. Weak nuclear force4. Gravity– Only gravity and electromagnetic forcesare relevant in classical mechanics( motion of macroscopic objects ).Decreasing strengthForce and acceleration• Larger force gives larger acceleration• Directly proportional:• But clearly different bodies acceleratedifferently under the same force.– Heavier objects are harder to push.– Proportionality constant may depend on weight?! a " FInertia again• But we already said that inertia characterizesa body’s tendency to retain its motion(I.e. to not change its velocity),We say a heavier object has more inertia.• But inertia and weight are different– A body in space is weightless,but it still resists a pushMass• Define mass to be‘the amount of inertia of an object’.• Can also say mass characterizes the amount ofmatter in an object.• Symbol for mass usually m• Unit of mass is the kilogram (kg).• Said before that• Find experimentally that! a " F ! Acceleration =ForceMass! a =FmForce, weight, and mass• 1 Newton = force required to accelerate a1!kg mass at 1 m/s2. ! F = ma " F = (kg)#(m/s2)= kg $ m / s2% NewtonBut then what is weight?—Weight is a force, measured in Newton’s—It is the net force of gravity on a body.—F=mg, g=F/m4What do you think?Suppose you are on the mooninstead of on earthA. Your weight is less but your mass is the same.B. Both your weight and mass are less than on earth.C. Your weight is less and your mass is zero.Mass is an intrinsic characteristic of a body.The force of gravity on the body (weight) willdepend on the other bodies around it.Is ‘pounds’ really weight?• In the English system (feet, pounds, seconds), poundsare a measure of force.• So it is correct to say my weight is 170 pounds.• Then what is my mass?! m =Fg=170lbs32 ft / s2= 5.3slugs!!Momentum conservation• We said before that an impressed force changesthe momentum of an object.• We also said that momentum is conserved.• This means the momentum of the object applyingthe force must have decreased.• According to Newton, there must be some forceacting on that object to cause the momentumchange.Newton’s third law• This is the basis for Newton’s third law:To every action there is always opposed anequal reaction.This is momentum conservation in thelanguage of forces.Colliding balls againBefore collision:After collision:1122During collision1 2Force on ball 2accelerates itForce on ball 1decelerates it tozero


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UW-Madison PHYSICS 107 - Momentum conservation, equal masses

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