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Chapter 6 Linear Momentum and Collisions September 20 Momentum and impulse

Learn about linear momentum, impulse, and the conservation of momentum in physics along with examples and applications in various scenarios. Understand the concept of elastic and inelastic collisions, their impact on kinetic energy and momentum conservation. Explore the importance of the center of mass in describing translational motion in objects.

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Chapter 6 Linear Momentum and Collisions September 20 Momentum and impulse

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  1. Chapter 6 Linear Momentum and Collisions September 20 Momentum and impulse 6.1 Linear Momentum Linear momentum: The linear momentum of an object is the product of its mass and velocity: It is often simply called momentum. It measures the quantity of motion of an object. Momentum is a vector. It is in the direction of the velocity of the object. Its component is The SI unit of momentum is kg·m/s. The total linear momentum of a system is the vector sum of the momenta of each individual objects: Example6.1: Momentum: Mass and Velocity Example6.3: Total Momentum: A Vector Sum

  2. The change in momentum is the vector difference between the final and initial momenta. Newton’s second law in terms of momentum: The net external force acting on an object is equal to the time rate of change of the momentum of the object. A change in momentum is evidence of a net force. Example: Projectile motion.

  3. 6.2 Impulse Impulse: The impulse on an object is the average force times the time duration of the force: Impulse is a vector, which is in the direction of the force. The SI unit of impulse is newton·second. It equals the unit of momentum: 1 N·s=1kg·m/s. Impulse-momentum theorem: The impulse exerted on an object equals to the change in the momentum of the object. If the force is not constant, the area under the F-vs-t curve equals to the impulse the object received, or the change in momentum of the object.

  4. Adjusting the collision force: Collision force can be reduced by increasing the contact time. Relation between kinetic energy and momentum: Example6.4: Teeing off: The Impulse–Momentum Theorem Example6.5: Impulse and Body Injury

  5. Read: Ch6: 1-2 Homework: Ch6: E3,5,20,31 Due: September 27

  6. September 23 Conservation of linear momentum 6.3 Conservation of Linear Momentum For a single object, For a system of objects, Law of conservation of linear momentum: The total linear momentum of a system is conserved if there is no net external force acting on the system. • Conservation of momentum is one of the most important principles in physics.In particular, it is used to analyze collisions and explosions of objects ranging from subatomic particles to automobiles in traffic accidents. • For an isolated system, by definition Fnet,ext= 0. • Conservation of momentum also applies to a coordinate component, e.g., if Fx-net,ext= 0 then Px is conserved.

  7. For an isolated system, if there are internal forces, the momenta of individual parts of the system may change, but the increase of the momentum of one parts is accompanied by the decrease of the momentum of other parts of the system, and the overall momentum of the system. stays the same. Example6.6: Before and After: Conservation of Momentum

  8. Example6.7: Conservation of Linear Momentum: Fragments and Components

  9. Read: Ch6: 3 Homework: Ch6: E38,46,48 Due: October 4

  10. September 24 Collisions 6.4 Elastic and Inelastic Collisions Collision is an interaction of objects that causes an exchange of energy and momentum. Elastic collision: In an elastic collision the total kinetic energy is conserved. That is, the total kinetic energy of all the objects in the system after the collision is the same as the total kinetic energy before the collision. Inelastic collision: In an inelastic collision, the total kinetic energy is not conserved. For example, one or more of the colliding objects may not regain the original shapes, or sound or frictional heat may be generated and some kinetic energy is lost. For an isolated system, momentum is conserved in both elastic and inelastic collisions. For an inelastic collision, only an amount of kinetic energy consistent with the conservation of momentum may be lost. Momentum is conserved because the forces of collision are internal to the system. The total kinetic energy may or may not be conserved because some kinetic energy may be changed into other forms of energy in the collision.

  11. Completely inelastic collision: In a completely inelastic collision the two objects stick together after collision. Example: One of the two objects (here m2) is initially at rest. Example6.9: Stuck Together: Completely Inelastic Collision

  12. Momentum and energy in elastic collisions: In an elastic collision both total momentum and total kinetic energy are conserved. Example: The general case of one-dimensional collision, final velocity v1, v2 =? Some special cases:

  13. Demo: Newton’s pendulum Example 6.10: Elastic Collision: Conservation of Momentum and Kinetic Energy Example6.11: Collisions: Overtaking and Coming Together Example6.12: Equal and Opposite

  14. Read: Ch6: 4 Homework: Ch6: E58,60,65 Due: October 4

  15. September 30 Center of mass 6.5 Center of Mass The center of mass (CM) is the point at which all of the mass of an object or system may be considered to be concentrated, for the purposes of describing its translational motion. Location of the center of mass of a system of particles: A rigid object is an object that does not change its shape. The relative position of the center of mass of a rigid object does not change. The relative position of the center of mass of a non-rigid object changes according to the instantaneous shape of the object. The center of mass of an object may be outside of the object, e.g., a moon-shape object.

  16. Total momentum of a system equals the momentum of its center of mass: (assuming all masses are concentrated at the center of mass) Newton’s second law for the motion of the center of mass of a system: Example: A diver’s center of mass Momentum conservation in terms of center of mass: If the net external force on a system is zero, the linear momentum of its center of mass is conserved. Example: A sliding wrench

  17. Example 6.14: Finding the Center of Mass: A Summation Process Example6.15: A Dumbbell: Center of Mass Revisited Example 6.16: Internal Motion: Where’s the Center of Mass and the Man? Center of gravity: The center of gravity of an object is the point at which all the weight of an object may be considered concentrated. It coincides with the center of mass when the acceleration due to gravity is constant. Demo: Center of gravity by suspension, and the flying of a board.

  18. Read: Ch6: 5 Homework: Ch6: E76,82 Due: October 9

  19. La nature ne s'est pas embarrassée des difficultés d'analyse. Nature is not embarrassed by difficulties of analysis. Augustin Fresnel

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