Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 1 Chapter 7: Linear Momentum Definition of Momentum Impulse Conservation of Momentum Center of Mass Motion of the Center of Mass Collisions (1d, 2d; elastic, inelastic)

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 2 § 7.1 Momentum Consider two interacting bodies: F 21 F 12 If we know the net force on each body then The velocity change for each mass will be different if the masses are different.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 3 Rewrite the previous result for each body as: Combine the two results:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 4 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.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 5 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.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 6 § 7.2 Impulse Definition of impulse: One can also define an average impulse when the force is variable.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 7 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 m 1 or m 2 Take m 1 <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.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 8 (b) Which mass has the greatest velocity change? Example continued: 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.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 9 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 =m  v.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 10 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.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 11 § 7.3 Conservation of Momentum v 1i v 2i m1m1 m2m2 m 1 >m 2 m1m1 m2m2 A short time later the masses collide. What happens?

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 12 During the interaction: N1N1 w1w1 F 21 N2N2 w2w2 F 12 x y There is no net external force on either mass.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 13 The forces F 12 and F 21 are internal forces. This means that: In other words, p i = p f. That is, momentum is conserved. This statement is valid during the interaction only.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 14 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? 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.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 15 § 7.4 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.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 16 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.)

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 17 In 3-dimensions write where The components of r cm are:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 18 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)? x y A x

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 19 Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 20 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? x y 3 2 1

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 21 Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 22 § 7.5 Motion of the Center of Mass For an extended body, it can be shown that p = mv cm. From this it follows that  F ext = ma cm.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 23 Example (text problem 7.35): Body A has a mass of 3.0 kg and v x = m/s. Body B has a mass of 4.0 kg and has v y = -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 m i. The position of each mass is r i and the displacement of each mass is  r i = v i  t.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 24 For the center of mass: Solving for the velocity of the center of mass: Or in component form: Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 25 Applying the previous formulas to the example, Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 26 §§ 7.6 & 7.7 Collisions in One and Two Dimension When there are no external forces present, the momentum of a system will remain unchanged. (p i = p f ) 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.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 27 If, after a collision, the bodies remain stuck together, the loss of kinetic energy is a maximum. This type of collision is called perfectly inelastic.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 28 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?

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 29 (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 1i = 1.0 m/s. Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 30 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?

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 31 Example (text problem 7.58): 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 v x =+1.0 m/s and v y =+2.0 m/s. What is the magnitude of body B’s velocity after the collision? A B v Ai Initial B A Final

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 32 x momentum: Solve for v 2fx : Solve for v 2fy : y momentum: The mag. of v 2 is Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 33 Summary Definition of Momentum Impulse Center of Mass Conservation of Momentum