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PHYS 172: Modern Mechanics Lecture 3 - The Momentum Principle Summer 2012 Read 2.1-2.6 The Momentum Principle An object moves in a straight line and at constant speed except to the extent that it interacts with other objects The Momentum Principle Dp = Fnet Dt Change of momentum is equal to the net force acting on an object times the duration of the interaction F units: N m kg/s2 • Fnet constant during t! What is “force” F? • measure of interaction. • defined by the momentum principle. The principle of superposition Fpush Dp = Fnet Dt Fearth Net force The Superposition Principle: The net force on an object is the vector sum of all the individual forces exerted on it by all other objects Each individual interaction is unaffected by the presence of other interacting objects Definition of net force: Fnet = F1 + F2 + ... Misconception: need constant force to maintain motion Fgravity Neglected friction! Impulse The Momentum Principle Definition of impulse Dp = Fnet Dt Impulse º Fnet Dt Note: small t Fnet ~ const Momentum principle: The change of the momentum of an object is equal to the net impulse applied to it Measuring force Fspring = kS s L0 L 1. Use knowledge of specific Interaction: Hooke’s Law, Gravitational Law, etc. Fgrav on 2by1 G m2 m1 r21 2 rˆ2 1 The Momentum Principle 2. Use the momentum principle Dp = Fnet Dt Predictions using the Momentum Principle The Momentum Principle Dp = Fnet Dt p f - pi = Fnet Dt p f = pi + Fnet Dt p fx , p fy , p fz pix , piy , piz Fnet , x , Fnet , y , Fnet , z t For components: p fx pix Fnet , x t p fy piy Fnet , y t p fz piz Fnet , z t Short enough, F~const Example Dp = Fnet Dt p f = pi + Fnet Dt Force: provided by a spring stretched by L=4 cm Find momentum pf if pi=<0,0,0> kg.m/s 1 second later 1. Force: Fspring = kS DL Fspring = 500(N/m)0.04(m)=20 N x NB: force must not change during t Fspring = 20,0,0 N 2. Momentum: p f = pi + Fnet Dt ( ) p f =< 0,0,0 > kg × m/s + 20,0,0 N × 1 s p f =< 20,0,0 > kg × m/s N.s = kg.m/s2.s = kg.m/s System and surroundings p system System: an object for which we calculate some property (force, momentum, etc) a system can consist of several objects Surroundings: objects which interact with system (earth, man, air…) p f = pi + Fnet Dt Only external forces matter ! Internal forces cancel Applying the Momentum Principle to a system: predicting motion 1. Choose system and surroundings 2. Make a list of objects in surroundings that exert significant forces on system 3. Apply the Momentum Principle p f = pi + Fnet Dt 4. Apply the position update formula if needed 5. Check for reasonableness (units, etc.) rf = ri + vavg Dt Example: a hockey puck A hockey puck with a mass of 0.16 kg is initially at rest. A player hits it applying force F 400, 400,0 N during t = 4 ms. Where would the puck be 2 seconds after it loses contact with hockey stick? Solution: 1. Choose a system and surroundings: Example: a hockey puck A hockey puck with a mass of 0.16 kg is initially at rest. A player hits it applying force F 400, 400,0 N during t = 4 ms. Where would the puck be 2 seconds after it loses contact with hockey stick? Solution: 1. Choose a system and surroundings: 2. Make a list of objects in surroundings that exert significant forces on system 3. Apply the Momentum Principle p f = pi + Fnet Dt p f = 0,0,0 ( m kg/s) + p f = 1.6,1.6,0 m× kg/s Hockey stick Earth Ice (floor) (normal force, ~friction) ( 400,400,0 N × 4×10-3s ) Example: a hockey puck A hockey puck with a mass of 0.16 kg is initially at rest. A player hits it applying force F 400, 400,0 N during t = 4 ms. Where would the puck be 2 seconds after it loses contact with hockey stick? Solution: y 3. Momentum pf p f = 1.6,1.6,0 m× kg/s x 4. The position update formula rf = ri + vavg Dt p » mv * Choose coordinate system origin: initial position of puck p v » = 10,10,0 m/s m ( ) rf = 0,0,0 m + 10,10,0 m/s× 2 s rf = 20,20,0 m rf Constant Gravitational Field Clicker question #2: g 9.8 N/kg (or m/s2 ) Due to the gravity, which components of the velocity will change? A) x, y and z B) only x and y C)only x D)only y E) only z y F 0, mg ,0 x z p fx pix Fnet , x t p fy piy Fnet , y t p fz piz Fnet , z t Shoot the monkey lecture demo Clicker question #5: The hunter aims right at the monkey and shoots. As the bullet leaves the rifle, the monkey sees the flash and releases the branch entering freefall. Will the bullet: A) Hit the monkey B) Undershoot the monkey C) Overshoot the monkey (ignore air resistance) Note: the trajectory is a parabola (if air friction is ignored)