Name_________________________ Period_______ Date______________ Ch. 9 OL: Center of Mass & Linear Momentum (p. 214—256) 9—1 Center of Mass (p. 214—219) 1. The center of mass ( ) of a system of particles is the point that moves as though… a) All of the system’s mass were… b) All external forces were… 2. The center of mass is the point where all of the mass of an object can be considered to be… 3. The center of mass is the “dot” that represents the object of interest in a… 4. The center of mass is defined to be the weighted average of the location of… ● Mathematically: xCM = = 5. In three dimensions, the center of mass must be identified by…Three coordinates ● ⃑ CM = 6. For more complex objects, the particles then become… 7. The sums of the equation above becomes the integrals of the coordinates of the… ● Mathematically: ⃑ CM = 8. Objects with non-uniform density have a linear density therefore… ● Mathematically: ⃑ CM ( ) given in mass per length, = 9. Solving for the mass you get… ● M = 10. The center of mass of an object need not lie within the… 11. Go to WileyPlus and click on the active figure voiceover. to watch Figure 9 - 4 animation with 12. Go to APlus Physics and watch this video on Center of Mass. The extra space is provided for notes. http://www.aplusphysics.com/courses/ap-c/videos/CenterOfMass/CenterOfMass.html. 13. A bar with a length of 30 cm has a linear density of λ = 10 + 6x, where x is in meters and λ is in kg/m. Determine the (a) mass of the bar and (b) the center of this bar. (a) 3.27 kg; (b) 0.154 m 9—2 Newton’s Second Law for a System of Particles (p. 220—223) 1. The vector equation that governs the motion of the center of mass of a system of particles is… ● = net 2. ⃑ net is the net force of all… 3. M is the total… 4. ⃑ CM is the acceleration of the… 5. Note that is equation in the same form as… ● ⃑ net = 6. We have determined the position of the center of mass: ● ⃑ CM = What about its velocity and acceleration? 7. We can calculate these quantities by differentiating the equation for ● ν CM = = = 8. Note that is the sum of the… = miri = r CM with respect to time: 9—3 Linear Momentum (p. 224—225) 1. We can say that an object as momentum ( ) if it is… 2. The linear momentum of a particle is a vector quantity ● ⃑ ( ) that is defined as… = 3. Since m is always a positive scalar quantity, and are in the… 4. The SI unit for momentum is the same as… ● kg m / s which is = 5. Newton expressed his second law of motion in terms of momentum… ● Mathematically: Fnet = = = = (m ν) 6. The linear momentum ⃑⃑ net of the system of particles is given by… ● ⃑⃑ net = which is another way to define the linear momentum of a system of particles: 7. The linear momentum of a system of particles is equal to the product of the total mass M of the system and the… 8. If we take the time derivative we find… ● ⃑ = = which give us Newton’s second law for a system of particles ● ⃑ net = 9—4 Collision and Impulse (p. 226—229) of any particle-like body cannot change unless a… 1. The momentum 2. An impulse ( ) is a vector quantity that describes the magnitude and duration of a… 3. Newton’s second law written in the form ⃑ net ● ⃑ net = = ∫ ⃑ dt can be integrated to find the net change in momentum = 4. Thus, the change in an object’s momentum is equal to the impulse on the object… ● = = ⃑ 5. The Impulse—Momentum theorem describes an unbalanced force that acts on an object for a period of time producing a… 6. If a graph of force-versus-time is given, then the impulse of force F as it acts from t1 to t2 is equal to the… 7. Go to APlus Physics and watch the video on Impulse & Momentum. Extra space is provided for notes. http://www.aplusphysics.com/courses/ap-c/videos/Momentum/Momentum.html. 8. During a collision with a wall lasting from t = 0 to t = 2 s, the force acting on a 2 kg object is given be the equation F = (4 kg m/s4) t (2s - t) ̂. (a) Calculate the impulse of the force on the object during the collision. (b) What is the average force on the object? (c) If the object starts from rest, what is the final velocity? (a) (c) ̂ m/s ̂ ; (b) ̂ N; 9—5 Conservation of Linear Momentum (p. 230—232) 1. The Law of Conservation of Linear Momentum states if no net external force acts on a system of particles, the…Total linear momentum ⃑ of the system cannot change ● Mathematically: ⃑ ⃑⃑ ⃑ = = 2. If the component of the net external force on a closed system is zero along an axis, then the component of the linear momentum of the system… 3. Although internal forces can change the linear momentum of portions of the system, they cannot change the… 4. The key to solving conservation of momentum problems is remembering that… 5. Go to APlus Physics and watch the video on Conservation of Momentum. http://www.aplusphysics.com/courses/honors/videos/ConsMomentum/ConsMomentum.html. Extra space is provided for notes. 9—6 Momentum and Kinetic Energy in Collisions (p. 233—236) 1. Collisions are classified into three categories: a. Elastic is where kinetic energy is… b. Inelastic is one in which the total kinetic energy is… i. The reason for the difference in total energy is due to the production of… 1. 2. 3. ii. If we know the values for, say, the masses, the initial velocities, and one of the final velocities, we can find the other… m1ν1i + m2ν2i = ν2f = c. Completely inelastic colliding objects stick together and move as… i. Note that V must be less than ν1i because of the greater… ii. V = 2. In a closed, isolated system, the velocity changed by a collision because… of the center of mass of the system cannot be 3. A two-body one-dimensional colliding system’s total linear momentum ⃑ equals… ⃑⃑ ● ⃑ = = = = 9—7 Elastic Collisions in One Dimension (p. 237—240) 1. In an elastic collision the total kinetic energy of the colliding bodies is conserved and is not… a. The net linear momentum is… m1ν1i ● = b. The total kinetic energy is… ½ m1 ● = 2. Combining the two above equations allows you to solve for the final velocity 1 & 2… ● ν1f = ● ν2f = 3. Note that 4. ν2f is always positive, that is it always… There are a few special situations to consider when objects collide... a. Equal masses: In head-on collisions, bodies of equal mass simply… ● ν1f = and ν2f = b. A massive target: The incoming smaller object simply… ● ν1f = and ν2f = c. A massive projectile: The incoming larger object simply keeps on going while the target’s velocity is… ● ν1f = d. Moving target: Setting and ν2i ν2f = = 0 gives you the same equation for ν1f & ν2f as in… 9—8 Collisions in Two Dimensions (p. 240—241) 1. When two bodies collide, the impulse between them determines the… 2. A glancing collision conserves both… 3. The key to solving a two-dimensional collision problem is to remember that momentum is a… 4. For components along the x axis… ● m1ν1i = 5. Components along the y axis… ● 0 = 6. In terms of kinetic energy… ● ½ m1 = 7. If we know any four of these quantities, we can solve for the three equations for the… 8. Go to APlus Physics and watch the video on Collisions in Multiple Dimensions. http://www.aplusphysics.com/courses/ap-c/videos/Collisions2D/Collisions2D.html. Extra space is provided for notes. 9. On a touchdown attempt, a 95 kg running back runs toward the end zone at 3.75 m/s. A 111 kg linebacker moving at 4.10 m/s meets the runner in a head-on collision. If the two players stick together, (a) what is their velocity immediately after the collision? (b) What are the initial and final kinetic energies of the system? (a) – 0.48 m/s; (b) 1600 J, 23.7 J