Chapter 7 Linear Momentum MFMc Graw Chap 07

Chapter 7 Linear Momentum MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010

Linear Momentum • Definition of Momentum • Impulse • Conservation of Momentum • Center of Mass • Motion of the Center of Mass • Collisions (1 d, 2 d; elastic, inelastic) MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 2

Momentum From Newton’s Laws Consider two interacting bodies with m 2 > m 1: m 1 F 21 F 12 m 2 If we know the net force on each body then The velocity change for each mass will be different if the masses are different. MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 3

Momentum Rewrite the previous result for each body as: From the 3 rd Law Combine the two results: MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 4

Define Momentum The quantity (mv) is called momentum (p). p = mv and is a vector. The unit of momentum is kg m/s; there is no derived unit for momentum. MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 5

Momentum From slide (4): 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. Δp 1 +Δp 2 = 0 MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 6

Impulse Definition of impulse: We use impulses to change the momentum of the system. One can also define an average impulse when the force is variable. MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 7

Impulse Example (text conceptual question 7. 2): A force of 30 N is applied for 5 sec to each of two bodies of different masses. 30 N Take m 1 < m 2 m 1 or m 2 (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. MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 8

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 a v the mass with the greater velocity change will have the greatest acceleration (mass 1). MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 9

Change in Momentum Example An object of mass 3. 0 kg is allowed to fall from rest under the force of gravity for 3. 4 seconds. Ignore air resistance. What is the change in momentum? Want p = m v. MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 10

Change in Momentum Example 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. MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 11

Conservation of Momentum v 1 i m 1>m 2 v 2 i m 1 m 2 A short time later the masses collide. m 1 m 2 What happens? MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 12

During the interaction: N 1 N 2 F 21 y F 12 w 1 x w 2 There is no net external force on either mass. MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 13

The forces F 12 and F 21 are internal forces. This means that: In other words, pi = pf. That is, momentum is conserved. This statement is valid during the interaction (collision) only. MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 14

Rifle Example 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? As long as the rifle is horizontal, there will be no net external force acting on the rifle-bullet system and momentum will be conserved. MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 15

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. MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 16

Center of Mass 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. ) MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 17

Center of Mass In 3 -dimensions write where The components of rcm are: MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 18

Center of Mass Example 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 MFMc. Graw x Chap 07 - Linear Momentum: Revised 3/3/2010 19

Center of Mass Example MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 20

Find the Center of Mass 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 1 x 3 MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 21

Example continued: MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 22

Motion of the Center of Mass For an extended body, it can be shown that p = mvcm. From this it follows that Fext = macm. MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 23

Center of Mass Velocity 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 = vi t. MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 24

Center of Mass Velocity For the center of mass: Solving for the velocity of the center of mass: Or in component form: MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 25

Center of Mass Velocity Applying the previous formulas to the example, MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 26

Collisions in One Dimension 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. MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 27

Collisions in One Dimension 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. MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 28

Inelastic Collisions in One Dimension 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. 0 m. The two cars couple together upon collision. (a) What is the speed of the two cars after the collision? MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 29

Inelastic Collisions in One Dimension (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 v 1 i = 1. 0 m/s. MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 30

One Dimension Projectile 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? MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 31

Collision in Two Dimensions Body A of mass M has an original velocity of 6. 0 m/s in the +x-direction toward a stationary body (B) of the same mass. After the collision, body A has vx = +1. 0 m/s and vy = +2. 0 m/s. What is the magnitude of body B’s velocity after the collision? Final Initial A A Angle is 90 o v. Ai B B MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 32

Collision in Two Dimensions x momentum: Solve for v 2 fx: y momentum: Solve for v 2 fy: The mag. of v 2 is MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 33

Summary • Definition of Momentum • Impulse • Center of Mass • Conservation of Momentum MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 34

Extra Slides MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 35

Conservation of Momentum Two objects of identical mass have a collision. Initially object 1 is traveling to the right with velocity v 1 = v 0. Initially object 2 is at rest v 2 = 0. After the collision: Case 1: v 1’ = 0 , v 2’ = v 0 (Elastic) Case 2: v 1’ = ½ v 0 , v 2’ = ½ v 0 (Inelastic) In both cases momentum is conserved. Case 1 MFMc. Graw Case 2 Chap 07 - Linear Momentum: Revised 3/3/2010 36

Conservation of Kinetic Energy? Case 1 KE After = KE Case 2 Before (Conserved) KE After = ½ KE Before (Not Conserved) KE = Kinetic Energy = ½mvo 2 MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 37

Where Does the KE Go? Case 1 KE After = KE Case 2 Before (Conserved) KE After = ½ KE Before (Not Conserved) In Case 2 each object shares the Total KE equally. Therefore each object has KE = 25% of the original KE Before MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 38

50% of the KE is Missing Each rectangle represents KE = ¼ KE MFMc. Graw Chap 07 - Linear Momentum: Revised 3/3/2010 Before 39

Turn Case 1 into Case 2 Each rectangle represents KE = ¼ KEBefore Take away 50% of KE. Now the total system KE is correct Use one half of the 50% taken to speed up object 1. Now it has 25% of the initial KE Use the other half of the 50% taken to slow down object 2. Now it has only 25% of the initial KE MFMc. Graw But object 2 has all the KE and object 1 has none Now they share the KE equally and we see where the missing 50% was spent. Chap 07 - Linear Momentum: Revised 3/3/2010 40