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Momentum and Impulse 8.01 W07D1 Fall 2006 Momentum and Impulse Obeys a conservation law Simplifies complicated motions Describes collisions Basis of rocket propulsion & space travel Quantity of Motion DEFINITION II Newton’s Principia The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjointly. The motion of the whole is the sum of the motions of all the parts; and therefore in a body double in quantity, with equal velocity, the motion is double; with twice the velocity, it is quadruple. Momentum and Impulse: Single Particle • Momentum r r p mv SI units [kg m s-1 ] [N s] • change in momentum • average force • impulse SI units r r p mv r r Fave p / t r r I Fave t [N s] tf r r I Fdt ti Impulse Demo To explain the demo and the example we need concepts of momentum and impulse An explanation is easier if we also use the idea of center of mass Jumping: Non-Constant Force • Plot of total external force vs. time for Andy jumping off the floor. Weight of Andy is 911 N. Jumping: Impulse • Shaded area represents impulse of total force acting on Andy as he jumps off the floor F t f 1.23 s I[ti ,t f ] ti 0.11 s total r r (t) N(t) Fgrav r total F (t) dt 199 N s Newton’s Second Law The change of motion is proportional to the motive force impresses, and is made in the direction of the right line in which that force is impressed, r r total dpr dv r F m ma dt dt tf r r total r r r I F dt p p(tf ) p(ti ) ti When then r dpr r F 0 dt r r p mv constant (vector) Concept Question: Impulse and Change in Momentum Identical constant forces push two identical objects A and B continuously from a starting line to a finish line. 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. Concept Question: Same Momentum, Different Masses Suppose a ping-pong ball and a bowling ball are rolling toward you. Both have the same momentum, and you exert the same force to stop each. How do the distances needed to stop them compare? 1. It takes a shorter distance to stop the ping-pong ball. 2. Both take the same distance. 3. It takes a longer distance to stop the ping-pong ball. Internal Force on a System of N Particles • The total internal force on the ith particle is sum of the interaction forces with all the other particles r Fint,i j N r Fi, j j 1 i j • The total internal force is the sum of the total internal force on each particle j N r total i1 r F Fint,i int i1 r Fi, j j 1 i j • Newton’s Third Law: internal forces cancel in pairs r r Fi, j Fj,i • So the total internal force is zero r total r Fint 0 Total Force on a System of N Particles is the External Force The total force on a system of particles is the sum of the total external and total internal forces. Since the total internal force is zero r total r total r total r total F Fext Fint Fext External Force and Momentum Change The total momentum of a system of N particles is defined as the sum of the individual momenta of the particles i N r r psys pi i1 Total force changes the momentum of the system r total r total F Fi i N i1 r r dpsys dp i dt i1 dt i N Total force equals total external force r total r total F Fext Newton’s Second and Third Laws for a system of particles: The total external force is equal to the change in momentum of the r system dp r total Fext sys dt Position and Velocity of Center of Mass • Total mass i N msys mi i1 • Position of center of mass r 1 i N r R cm mi ri msys i1 • Velocity of center of mass r i N p r 1 r sys Vcm m v msys i1 i i msys Translational Motion of the Center of Mass of Rigid Body • Momentum of system r r psys msys Vcm • System can be treated as point mass located at center of mass. External force accelerates center of mass r r r dpsys r total dVcm Fext msys msys Acm dt dt • Impulse changes center of mass momentum tf r r total r r r I Fext dt psys msys ( Vcm (tf ) Vcm (ti )) ti Jumping: Center of Mass Velocity • Plot of center of mass velocity vs. time starting from rest Vcm, y (ti ) 0 • Impulse for time interval [ti, t] equals change in momentum of system. t I y [ti ,t] Fytotal (t) dt Psys,y m(Vcm, y (t) Vcm, y (ti )) mVcm, y (t) ti 0.11 s • Center of mass velocity at time t is then Vcm, y (t) I y [ti ,t] / m Jumping: Center of Mass Velocity • When Andy leaves the ground, the impulse is I y [0.11 s,1.23 s] 199 N s • So the center of mass velocity is Vcm, y (t) I y [ti ,t] / m (199 N s)(9.80 m s-2 )/(911 N) 2.14 m s-1 What was the magnitude of the displacement of Andy’s center of mass after he left the floor? 2 Ycm Vcm,y (t f ) / 2g (2.14 m s-1 )2 / (2)(9.80 m s-2 ) 0.23 m What is the shape of the trajectory of the center of mass of this coin? CM 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 field Fosbury Flop Concept Question: Pushing Baseball Bat 1 2 3 The greatest acceleration of the center of mass will be produced by pushing with a force F at 1. Position 1 2. Position 2 3. Position 3 4. All the same