PAP111_Lecture13 - PAP 111 Mechanics and Relativity Lecture...

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Unformatted text preview: PAP 111 Mechanics and Relativity Lecture 13 Angular Momentum Angular Momentum and Torque Angular Momentum of a Rotating Rigid Object Conservation of Angular Momentum The Motion of Gyroscopes and Tops Revision on Vector Cross Product Given two vectors A and B , their vector cross product is defined by: where the magnitude of C is AB sin . A and B are the magnitude of the vector A and B respectively, while is the smaller angle between A and B . B A C = Note that the quantity AB sin is equal to the area of the parallelogram formed by A and B . The direction of C is perpendicular to the plane formed by A and B . The best way to determine the direction of C is to use the right-hand rule. Properties of Vector Cross Product The vector product is not commutative, i.e., . If A is parallel to B ( =0 or 180), then . If A is perpendicular to B , then . The vector product obeys the distributive law: The derivative of the cross product with respect to a variable t is A-B B A = B A = AB = B A ( 29 C A B A C B A + = + ( 29 dt d dt d dt d B A B A B A + = Vector Products from Unit Vectors The vector products between the Cartesian unit vectors , and are given by The cross product can be expressed as k j i j k i- i k i j k- k j k i j- j i k k j i i i = = = = = = = = = ( 29 ( 29 ( 29 k j- i k j- i k j i B A x y y x x z z x y z z y y x y x z x z x z y z y z y x z y x B A B A B A B A B A B A B B A A B B A A B B A A B B B A A A- +-- = + = = Formal Definition of Torque The torque lies in a direction perpendicular to the plane formed by the position vector r and the applied force vector F . More formally, it is defined as follows: F r = (15.1) Angular Momentum - Preliminary Consider a particle of mass m located at the vector position r and moving with linear momentum p . If this particle is acted on by a net force F , we can write = = dt d p r F r ( 29 dt d dt d dt d p r p r p r = + = Let us add to the RHS the term d r /d t x p , which is zero since d r /d t = v , and v and p are parallel. Thus, The above equation is similar in form to F =d p /d t . This suggests that r x p plays the same role in rotational motion as p plays in translational motion. Angular Momentum...
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PAP111_Lecture13 - PAP 111 Mechanics and Relativity Lecture...

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