PAP111_Lecture13 - PAP111 Lecture13 AngularMomentum Product...

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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
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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 right-hand  rule.
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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 A -B B A × = × 0 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 × + × = ×
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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 0 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 - + - - = + = = ×
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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)
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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 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.
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Angular Momentum The instantaneous  angular momentum L  of a 
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