Momentum and Impulse 8.01 W05D1 Today’s Reading Assignment (W05D1): MIT 8.01 Course Notes Chapter 10 Momentum, System of Particles, and Conservation of Momentum Sections 10.1-10.9Announcements Problem Set 3 due Week 5 Tuesday at 9 pm in box outside 26-152 Math Review Week 5 Tuesday 9-11 pm in 26-152. Add Date Friday Oct 4Learning Resources and Suggestions Sunday Tutoring Session 1-5 pm in 26-152 Office Hours: You can attend any office hour that fits your schedule Join Seminar XL Study Group TSR Tutorial Service Room 12-124: Individual tutoring Physics Homework Nights Sunday in 12-124 from 7 to 11 p.m. Study Suggestions: Print Today’s Presentation Slides PDF (Print) version and bring to each class Print and work through In-class and Friday Problem Solving solutions Print and work through Problem Set solutions if you had trouble with any particular problemMomentum and Impulse Obeys a conservation law Simplifies complicated motions Describes collisions Basis of rocket propulsion & space travelMomentum and Impulse: Single Particle • Momentum SI units • Change in momentum • Force • Impulse • SI units m=p vmΔ = Δp v [kg ⋅ m ⋅ s-1] = [N ⋅ s] [N ⋅ s] F = ma = mdvdt=dpdt I ≡F dttitf∫=dpdtdttitf∫= dptitf∫= Δp ≡p(tf) −p(ti)Concept Question: Pushing Identical Objects Identical constant forces push two identical objects A and B continuously from a starting line to a finish line. Neglect friction. If A is initially at rest and B is initially moving to the right, 1. Object A has the larger change in momentum. 2. Object B has the larger change in momentum. 3. Both objects have the same change in momentum 4. Not enough information is given to decide.Table Problem: Impulse and Superball A superball of m, starting at rest, is dropped from a height hi above the ground and bounces back up to a height of hf. The collision with the ground occurs over a time interval Δtc. Ignore air resistance. a) What is the momentum of the ball immediately before the collision? b) What is the momentum of the ball immediately after the collision? c) What impulse is imparted to the ball? d) What is the average force of the ground on the ball?Concept Question: Impulse The figure to the right depicts the paths of two colliding steel balls, A and B. Which of the arrows 1-5 best represents the impulse applied to ball B by ball A during the collision?Demo: Jumping Off the Floor with a Non-Constant ForceDemo Jumping: Non-Constant Force • Plot of total external force vs. time for Andy jumping off the floor. Weight of Andy is 911 N.Demo Jumping: Impulse • Shaded area represents impulse of total force acting on Andy as he jumps off the floor 1.23 stotal0.11s[ , ] ( ) 199 N sfiti ftt t t dt=== = ⋅∫I F total( ) ( )gravt t= +F N F Demo Jumping: Height When Andy leaves the ground, the impulse is So the y-component of his velocity is Andy jumped Iy[0.11 s,1.23 s] = 199 N ⋅ svy, f= Iy[ti,tf] / m = (199 N ⋅ s)(9.80 m ⋅ s-2)/(911 N) = 2.14 m ⋅ s-1(1 / 2)mvy, f2= mgh ⇒ h = vy, f2/ 2g = 23.3 cmSystem of Particles: Center of MassPosition and Velocity of Center of Mass Mass for collection of discrete bodies (system): Momentum of system: Position of center of mass Velocity of center of mass Rcm=1msysmirii =1i = N∑ Vcm=1msysmivii =1i = N∑=psysmsys⇒ msys= mii =1i = N∑ psys= mivii =1i = N∑ psys= msysVcmContinuous Bodies of Center of Mass Infinitesimal mass element Mass for system: Position vector for infinitesimal element Position of center of mass Velocity vector for infinitesimal element Velocity of center of mass ri→r Vcm=1msysmivii=1i= N∑→1msysdmv =body∫psysmsys msys= mii=1i= N∑→ dmbody∫ mi→ dm =ρdV , volume elementσdA , area elementλds , length element⎧⎨⎪⎩⎪ Rcm=1msysmirii=1i= N∑→1msysdmrbody∫ vii→vTable Problem: Center of Mass of Rod and Particle Post- Collision A slender uniform rod of length d and mass m rests along the x-axis on a frictionless, horizontal table. A particle of equal mass m is moving along the x-axis at a speed v0. At t = 0, the particle strikes the end of the rod and sticks to it. Find a vector expression for the position of the center of mass of the system for (i) t = 0, (ii) t > 0. The figure on the right shows an overhead view of the rod lying on the table.System of Particles: Internal and External Forces, Center of Mass MotionSystem of Particles: Newton’s Second and Third Laws The momentum of a system remains constant unless the system is acted on by an external force in which case the acceleration of center of mass satisfies Fext=dpsysdt= msysdVcmdt= msysAcmDemo : Center of Mass Trajectory B78 http://tsgphysics.mit.edu/front/?page=demo.php&letnum=B 78&show=0 Odd-shaped objects with their centers of mass marked are thrown. The centers of mass travel in a smooth parabola. The objects consist of: a squash racket, a 16” diameter disk weighted at one point on its outer rim, and two balls connected with a rod. This demonstration is shown with UV light. Video link: http://techtv.mit.edu/videos/3052-center-of-mass-trajectoryCM moves as though all external forces on the system act on the CM so the jumper’s cm follows a parabolic trajectory of a point moving in a uniform gravitational fieldCenter of mass passes under the barTable Problem: Exploding Projectile Center of Mass Motion An instrument-carrying projectile of mass m1 accidentally explodes at the top of its trajectory. The horizontal distance between launch point and the explosion is x0. The projectile breaks into two pieces which fly apart horizontally. The larger piece, m3, has three times the mass of the smaller piece, m2. To the surprise of the scientist in charge, the smaller piece returns to earth at the launching station. a) How far has the center of mass of the system traveled from the launch when the pieces hit the ground? b) How far from the launch point has the larger piece traveled when it first hits the ground?Internal Force on a System of N Particles is Zero • The internal force on the ith particle is sum of the interaction forces with all the other particles • The internal force is the sum of the internal force on each particle •
View Full Document
Unlocking...