VIEWS: 12 PAGES: 66 POSTED ON: 9/8/2011 Public Domain
Momentum • Do the Yankees or the Phillies have Big Mo (momentum) with them tonight? • If you have ever been in an auto accident physics may have saved your life or at least prevented more serious injuries. • Conservation of Momentum as a forensic tool. Momentum Inertia in motion • Inertia is a measure of the resistance of an object to a change in motion. • We have a sense that it is harder to stop a truck than a car if both are moving at the same speed. • It is more difficult to stop a car moving quickly than one that is moving slowly. Momentum • Momentum = mass x velocity • =mv • Momentum is a vector • We often write momentum as p, p =mv • The unit of momentum is kg m/s; there is no derived unit for momentum. Momentum • An object can have a large momentum if • m is large, mv , or if v • It has a large speed, m , or • Both • Can a skateboard have more momentum than a cement truck? A) Yes B) No Impulse • To change momentum we must change either mass or velocity (or both). • If m is constant to change p we must change v (accelerates) • Δp = mΔv • What causes a? A force F. • But there is another variable when changing momentum, time! Impulse • If you are pushing a car without gas the final velocity depends on both how hard and how long you push. • The final momentum thus depends on how large a force and on how long you apply the force. • Definition of impulse • Can also define an average impulse when force is variable Impulse-examples Increase momentum (increase Δt) • Follow through- golf, baseball • Barrel of rifle Decrease momentum Increase Δt to decrease F Impulse • The concept of impulse leads to a more general form of Newton’s 2nd law. • FΔt = Δp • F = Δp/Δt • In this form we can handle problems where the mass changes • If m is constant Δp = mΔv • F = m Δv/Δt = ma Example (text problem 7.9): An object of mass 3.0 kg is allowed to fall from rest under the force of gravity for 3.4 seconds. What is the change in momentum? Ignore air resistance. Want p = mv. 12 Example : A force of 30 N is applied for 5 sec to each of two bodies of different masses. 30 N Take m1 < m2 m1 or m2 (a) Which mass has the greater momentum change? Since the same force is applied to each mass for the same interval, p is the same for both masses. 13 Example continued: (b) Which mass has the greatest velocity change? Since both masses have the same p, the smaller mass (mass 1) will have the larger change in velocity. (c) Which mass has the greatest acceleration? Since av the mass with the greater velocity change will have the greatest acceleration (mass 1). 14 Example (text problem 7.10): What average force is necessary to bring a 50.0-kg sled from rest to 3.0 m/s in a period of 20.0 seconds? Assume frictionless ice. The force will be in the direction of motion. 15 Question As a linebacker on the football team, which player would be the easiest for you to stop when you tackle him? A) The 350-lb lineman who can cover 10 yards in 2.5 seconds. B) The 160-lb running back, who can make it down the field in ten seconds. C) The 240-lb halfback, who can make it down the field in twenty seconds. D) They are all equally hard to stop. Question As a kid playing on the playground, you would bend your knees when you landed after jumping off the monkey bars to reduce the "sting" in your feet. This worked because A) bending your knees gave you upward momentum which partly canceled the downward momentum. B) bending your knees lowered your center of gravity reducing the force of your fall. C) bending your knees increased the time of contact for the ground to bring you to rest. D) you didn't do it on purpose, your knees just buckled. Momentum Consider two interacting bodies with m2 > m1: F21 F12 m1 m2 If we know the net force on each body then The velocity change for each mass will be different if the masses are different. 18 Rewrite the previous result for each body as: Newton’s 3rd Law Combine the two results: 19 From slide (3): The change in momentum of the two bodies is “equal and opposite”. Total momentum is conserved during the interaction; the momentum lost by one body is gained by the other. 20 Conservation of Momentum • If the net external force acting on a system is zero, then the momentum of the system does not change. • When a quantity in physics does not change we say it is conserved. Momentum is conserved. • We have to be careful to define the system. • It is only forces external to the system that can change momentum Conservation of Momentum • Internal forces do not change the momentum of the system. Momentum is a vector. Each component is conserved separately. Vectors can cancel each other out. 0 -M v + m V = 0 Conservation of Momentum Crash! What happens in collisions Energy and Momentum in collisions. Conservation of Momentum • If the net external force acting on a system is zero, then the momentum of the system does not change. • We have to be careful to define the system. • It is only forces external to the system that can change momentum. • If there are no external forces to the system the internal changes in momentum cancel Question You peddle frantically to get your bicycle up to a speed of 15 m/s. On level ground, you relax and start to coast. Your speed A) stays the same, because momentum is conserved. B) increases, now that you are not expending energy turning the pedals. C) decreases, as there are still forces acting on the bicycle. Conservation of Momentum v1i v2i m1>m2 m1 m2 A short time later the masses collide. m1 m2 What happens? 29 During the interaction: N1 N2 y F21 F12 x w1 w2 There is no net external force on either mass. 30 The forces F12 and F21 are internal forces. This means that: In other words, pi = pf. That is, momentum is conserved. This statement is valid during the interaction only. 31 Example (text problem 7.18): A rifle has a mass of 4.5 kg and it fires a bullet of 10.0 grams at a muzzle speed of 820 m/s. What is the recoil speed of the rifle as the bullet leaves the barrel? 32 Question You paddle your own canoe forward by pushing back on the water. If you can change the velocity of 7.3 kg of water by 3.0 m/s with each stroke, your speed changes by (myou + mcanoe = 93 kg) A) 226 m/s B) 38 m/s C) 4.2 m/s D) 0.24 m/s Collisions When there are no external forces present, the momentum of a system will remain unchanged. (pi = pf) If the kinetic energy before and after an interaction is the same, the “collision” is said to be perfectly elastic. If the kinetic energy changes, the collision is inelastic. 34 Elastic collisions Inelastic collisions If, after a collision, the bodies remain stuck together, the loss of kinetic energy is a maximum, but not necessarily a 100% loss of kinetic energy. This type of collision is called perfectly inelastic. 37 Example (text problem 7.41): In a railroad freight yard, an empty freight car of mass m rolls along a straight level track at 1.0 m/s and collides with an initially stationary, fully loaded, boxcar of mass 4.0m. The two cars couple together upon collision. (a) What is the speed of the two cars after the collision? 39 (b) Suppose instead that both cars are at rest after the collision. With what speed was the loaded boxcar moving before the collision if the empty one had v1i = 1.0 m/s. 40 Example (text problem 7.49): A projectile of 1.0 kg mass approaches a stationary body of 5.0 kg mass at 10.0 m/s and, after colliding, rebounds in the reverse direction along the same line with a speed of 5.0 m/s. What is the speed of the 5.0 kg mass after the collision? 41 Question Which is an example of a perfectly elastic collision? A) A wet tissue thrown at the wall. B) The cue ball striking the 8-ball in a billiards game. C) A rubber ball bouncing off the floor. Fig. 07.18 The New York Times, Jan.13, 1920, p. 12 …its flight would be neither accelerated nor maintained by the explosion of the charges… To claim that it would be, is to deny a fundamental law of dynamics… That Professor Goddard, with his ‘chair’ in Clark College and the countenancing of the Smithsonian Institution, does not know the relation of action to reaction, and of the need to have something better than a vacuum against which to react – to say that would be absurd. PHY211 Fall 2009 Lecture 10-1 The New York Times, July 17, 1969, p. 34 …further investigation and experimentation have confirmed the findings of Isaac Newton in the 17th century, and it is now definitely established that a rocket can function in a vacuum as well as in an atmosphere. The Times regrets the error. …an editorial feature of the New York Times dismissed the notion that a rocket could function in a vacuum and commented on the ideas of Robert H. Goddard. PHY211 Fall 2009 Lecture 10-1 Question A boxcar with mass m moving to the right with speed v collides with a second boxcar of the same mass which is at rest. After the collision, both boxcars move off to the right with speed v/2. In this collision A) momentum is conserved. B) energy is conserved. C) momentum and energy are conserved. D) neither momentum nor energy is conserved. Center of Mass • Why is your belly button where it is? • Something simple about complicated motion • Why can I safely walk out over the edge of the table? There is something special about this point Page 250 §7.5 Center of Mass The center of mass (CM) is the point representing the mean (average) position of the matter in a body. This point need not be located within the body. 54 The center of mass (of a two body system) is found from: This is a “weighted” average of the positions of the particles that compose a body. A larger mass is more important. If m1 = 100 m2 then xcm ~ x1 55 In 3-dimensions write where The components of rcm are: 56 Example (text problem 7.30): The positions of three particles are (4.0 m, 0.0 m), (2.0 m, 4.0 m), and (1.0 m, 2.0 m). The masses are 4.0 kg, 6.0 kg, and 3.0 kg respectively. What is the location of the center of mass? y 2 M r 4 (4,0) 1 x 6 (2,4) 3 3 (-1,-2) 57 Example continued: 58 Example (text problem 7.27): Particle A is at the origin and has a mass of 30.0 grams. Particle B has a mass of 10.0 grams. Where must particle B be located so that the center of mass (marked with a red x) is located at the point (2.0 cm, 5.0 cm)? y x A x 59 Where is the CM of a Donut? A B C D) It could be A or C or anywhere on the donut. Motion of the Center of Mass For an extended body, it can be shown that p = mvcm. From this it follows that Fext = macm. 62 Fig. 07.11 Example (text problem 7.35): Body A has a mass of 3.0 kg and vx = +14.0 m/s. Body B has a mass of 4.0 kg and has vy = 7.0 m/s. What is the velocity of the center of mass of the two bodies? Consider a body made up of many different masses each with a mass mi. The position of each mass is ri and the displacement of each mass is ri = vit. 64 Example continued: For the center of mass: Solving for the velocity of the center of mass: Or in component form: 65 Example continued: Applying the previous formulas to the example, 66