Conservation of angular momentum
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Conservation of Angular Momentum
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When two objects are involved in a head-on collision we believe that their total linear momentum is conserved - this is the substance of Newton's famous second law. The momentum of the combined objects before the collision is equal to the moment of the combined objects after the collision. Forces not only propel objects in straight lines; they also cause them to rotate. Rotating objects have angular momentum. So we would expect the angular momentum of objects involved in a collision also to be conserved - that is, the sum of the angular momentums of the particles before a collision is equal to the sum of the angular momentums of those particles after collision. Thus we seek an analogy to the familiar result that in any closed system linear momentum is conserved for angular momentum. That analog is the conservation of angular momentum: the angular momentum of any closed system that is rotating is conserved - that is, does not change. By analogy with linear momentum, angular momentum, is defined to be: angular momentum = moment of inertia ´ angular velocity [Equation goes here - download the original to see it.] By analogy with conservation of linear momentum, conservation of angular momentum entails that, for any closed system, the sum of all the angular momentums of the constituent parts is always constant. [Equation goes here - download the original to see it.] Angular momentum is also called moment of momentum. If a system is subject to an external torque (also called moment or couple) then the angular momentum of such as system will not be conserved. Such a system is said to be open. In an open system angular momentum is not conserved Firstly, we illustrate the application of conservation of momentum; then we demonstrate the consistency of the definition of angular momentum and prove the theorem that in a closed system angular momentum is always conserved. Example (1) An ice-skater rotates about a fixed vertical axis with moment of inertia 6.0 Kg m2 when her arms are extended. She draws her arms to her side, and her moment of inertia changes to 2.4 Kg m2. Given that her initial angular velocity is 7.2 rads-1 find her final angular velocity once she has brought her arms down by her side. Conservation of angular momentum gives [Equation goes here - download the original to see it.] We now demonstrate the consistency of the definition of angular momentum with the definition of linear momentum. Let P be a particle of mass M rotating about an axis L. Let r be the perpendicular distance of P from L. [Diagram goes here - download the original to see it.] As usual the tangental velocity is [Equation goes here - download the original to see it.] Then the tangental momentum is [Equation goes here - download the original to see it.] Note that since P is in orbit there is no radial component of the linear momentum, so the tangental linear momentum is all the momentum there is. Now the moment of a force - its torque - is defined to be: moment = force ´ perpendicular distance C [Equation goes here - download the original to see it.] = Fr.By analogy, the moment of the linear momentum (called moment of momentum or angular momentum) is moment of momentum = linear momentum x perpendicular distance [Equation goes here - download the original to see it.]
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Contents of
Conservation of angular momentum
1 Conservation of Angular Momentum
2 Proof of the result (conservation of angular momentum)
3 Rotational kinetic energy
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