This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 θ . A and B are the magnitude of the vector A and B respectively, while θ is the smaller angle between A and B . B A C × = Note that the quantity AB sin θ 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 righthand rule. Properties of Vector Cross Product The vector product is not commutative, i.e., . If A is parallel to B ( θ =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 AB 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 ˆ j ˆ i ˆ j k i i k i j k k j k i j j i k k j i i i ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ = × = × = × = × = × = × = × = × = × ( 29 ( 29 ( 29 k j i k j i k j i B A ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 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 τ 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 × = τ (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 Σ F , we can write ∑ × = = ∑ × dt d p r F r τ ( 29 dt d dt d dt d p r p r p r × = × + ∑ × = τ 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 Σ 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...
View
Full
Document
This note was uploaded on 12/08/2010 for the course SPMS pap111 taught by Professor Clausdieterohl during the Fall '10 term at Nanyang Technological University.
 Fall '10
 ClausDieterOhl

Click to edit the document details