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# Conservation of Momentum

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PPT on concepts related to conservation of Momentum for Class XI

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### Conservation of Momentum

1. 1. Momentum for Class-XI By Yatendra Kumar
2. 2. Momentum defined <ul><li>Objects in motion are said to have a momentum. </li></ul><ul><li>This momentum is a vector. </li></ul><ul><ul><li>It has a size and a direction. </li></ul></ul><ul><ul><li>The size of the momentum is equal to the mass of the object multiplied by the size of the object's velocity. </li></ul></ul><ul><ul><li>The direction of the momentum is the same as the direction of the object's velocity. </li></ul></ul>
3. 3. Difference between KE and Momentum kinetic energy … momentum … is a scalar. is a vector is not changed by a force perpendicular to the motion, which changes only the direction of the velocity vector. is changed by any force, since a change in either the magnitude or the direction of the velocity vector will result in a change in the momentum vector. is always positive, and cannot cancel out. cancels with momentum in the opposite direction. can be traded for other forms of energy that do not involve motion. KE is not a conserved quantity by itself. is always conserved in a closed system. is quadrupled if the velocity is doubled. is doubled if the velocity is doubled.
4. 4. classical mechanics Vs relativistic mechanics <ul><li>In classical mechanics , momentum (is the product of the mass and velocity of an object ( p  =  m v ). </li></ul><ul><li>In relativistic mechanics , this quantity is multiplied by the Lorentz factor . </li></ul><ul><li>http://en.wikipedia.org/wiki/Momentum </li></ul>
5. 5. Linear Vs Angular Momentum <ul><li>Momentum is sometimes referred to as linear momentum to distinguish it from the related subject of angular momentum . </li></ul><ul><li>Linear momentum is a vector quantity, since it has a direction as well as a magnitude. </li></ul><ul><li>Angular momentum is a pseudovector quantity because it gains an additional sign flip under an improper rotation . </li></ul>
6. 6. Linear Momentum <ul><li>If an object is moving in any reference frame , then it has momentum in that frame. </li></ul><ul><li>It is important to note that momentum is frame dependent . </li></ul><ul><li>That is, the same object may have a certain momentum in one frame of reference, but a different amount in another frame. </li></ul><ul><ul><li>For example, a moving object has momentum in a reference frame fixed to a spot on the ground, while at the same time having 0 momentum in a reference frame attached to the object's center of mass . </li></ul></ul><ul><li>The amount of momentum that an object has depends on two physical quantities: the mass and the velocity of the moving object in the frame of reference . </li></ul><ul><li>In physics, the usual symbol for momentum is a bold p (bold because it is a vector ); so this can be written </li></ul><ul><li>where p is the momentum, m is the mass and v is the velocity. </li></ul>
7. 7. Momentum and frame of reference <ul><li>Newton's apple in Einstein's elevator. </li></ul><ul><li>In person A's frame of reference, the apple has non-zero velocity and momentum. </li></ul><ul><li>In the elevator's and person B's frames of reference, it has zero velocity and momentum. </li></ul>
8. 8. Linear momentum of a system of particles <ul><li>Relating to mass and velocity </li></ul><ul><li>The linear momentum of a system of particles is the vector sum of the momenta of all the individual objects in the system: </li></ul><ul><li>where P is the total momentum of the particle system, m i and v i are the mass and the velocity vector of the i -th object, and n is the number of objects in the system. </li></ul><ul><li>In the center of mass frame the momentum of a system is zero. </li></ul><ul><li>Momentum with respact to frame of reference in which the Centre of Mass is moving with velocity Vcm the momentum will be </li></ul><ul><li>where </li></ul>.
9. 9. Conservation of linear momentum <ul><li>The law of conservation of linear momentum is a fundamental law of nature, and it states that the total momentum of a closed system of objects (which has no interactions with external agents) is constant. </li></ul><ul><li>One of the consequences of this is that the center of mass of any system of objects will always continue with the same velocity unless acted on by a force from outside the system. </li></ul><ul><li>In an isolated system with only two objects, the change in momentum of one object must be equal and opposite to the change in momentum of the other object. </li></ul>
10. 10. Conservation within a system not experiencing external force <ul><li>If you have several objects in a system, perhaps interacting with each other, </li></ul><ul><ul><li>but not being influenced by forces from outside of the system, then the total momentum of the system does not change over time. </li></ul></ul><ul><li>However, the separate momenta of each object within the system may change. </li></ul><ul><li>One object might change momentum, say losing some momentum, as another object changes momentum in an opposite manner, picking up the momentum that was lost by the first. </li></ul>
11. 11. Conservation of momentum <ul><li>The total momentum of any group of objects remains the same unless outside forces act on the objects (law of conservation of momentum ). </li></ul><ul><li>Hence two objects colliding each other will have their momentum conserved as no external force is acting on them </li></ul>
12. 12. Common problem <ul><li>A common problem in physics that requires the use of this fact is the collision of two particles. </li></ul><ul><li>Since momentum is always conserved, the sum of the momenta before the collision must equal the sum of the momenta after the collision: </li></ul>
13. 13. Conservation law-wider implication <ul><li>Momentum is a conserved quantity, meaning that the total momentum of any closed system (one not affected by external forces) cannot change. </li></ul><ul><li>Although originally expressed in Newton's Second Law , it also holds in special relativity , and with appropriate definitions a (generalized) momentum conservation law holds in electrodynamics , quantum mechanics , quantum field theory , and general relativity . </li></ul>
14. 14. Understanding Newton’s second law for conservation of momentum <ul><li>The second law states that the net force on a particle is equal to the time rate of change of its linear momentum p in an inertial reference frame : </li></ul><ul><li>This is valid only for constant mass system. </li></ul>
15. 15. When net Force is zero <ul><li>As </li></ul><ul><li>When F=0, that there is no external force, the momentum remains conserved </li></ul>
16. 16. Elastic Collision <ul><li>In elastic collision both kinetic energy and momentum are conserved. Provided there is no external force </li></ul><ul><li>In inelastic collision, only momentum is conserved. The Kinetic energy is not conserved. It may be lost as heat, sound ,etc. </li></ul>
17. 17. Elastic collision in one dimension <ul><li>Momentum is conserved </li></ul><ul><li>Kinetic energy is also conserved- </li></ul><ul><li>The equations for the velocities after collision </li></ul>
18. 18. Two bodies with equal mass-perfectly elastic collision <ul><li>Head-on collision of two bodies of equal masses. </li></ul><ul><li>A Newton's cradle demonstrates conservation of momentum. </li></ul>
19. 19. Perfectly inelastic collisions <ul><li>In perfectly in elastic collision, the two bodies collide and stick together. </li></ul><ul><li>A common example of a perfectly inelastic collision is when two snowballs collide and then stick together afterwards. This equation describes the conservation of momentum: </li></ul><ul><li>It can be shown that a perfectly inelastic collision is one in which the maximum amount of kinetic energy is converted into other forms. </li></ul><ul><li>For instance, if both objects stick together after the collision and move with a final common velocity, one can always find a reference frame in which the objects are brought to rest by the collision and 100% of the kinetic energy is converted. </li></ul><ul><li>This is true even in the relativistic case and utilized in particle accelerators to efficiently convert kinetic energy into new forms of mass-energy (i.e. to create massive particles). </li></ul>
20. 20. Coefficient of restitution and collision <ul><li>The coefficient of restitution is defined as the ratio of relative velocity of separation to relative velocity of approach. </li></ul><ul><li>It is a ratio hence it is a dimensionless quantity. </li></ul><ul><li>A perfectly elastic collision implies that C R is 1. </li></ul><ul><ul><li>So the relative velocity of approach is same as the relative velocity of separation of the colliding bodies. </li></ul></ul><ul><li>Inelastic collisions have (C R < 1). </li></ul><ul><li>In case of a perfectly inelastic collision C R is 0 </li></ul>
21. 21. Understanding impulse <ul><li>An impulse I occurs when a force F acts over an interval of time Δ t , and it is given by </li></ul><ul><li>Since force is the time derivative of momentum with m being constant, it follows that </li></ul><ul><li>This relation between impulse and momentum is closer to Newton's wording of the second law. </li></ul>
22. 22. Source <ul><li>http://zonalandeducation.com/mstm/physics/mechanics/momentum/momentum.html </li></ul><ul><li>http://en.wikipedia.org/wiki/Momentum </li></ul><ul><li>http://tap.iop.org/mechanics/momentum/page_39569.html </li></ul><ul><li>http://www.lightandmatter.com/html_books/2cl/ch04/ch04.html </li></ul>