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Unformatted text preview: Angular Momenta and Torques Shina Tan Angular momentum and torque about any fixed point By a fixed point we mean a point that is at rest in the given coordinate system. Definition 1 The angular momentum of a system of particles about any fixed reference point o is defined as L ( o ) ( t ) = X [ r ( t )- o ] m r ( t ) , (1) where [ r ( t )- o ] is the position vector of the -th particle with respect to the reference point. Definition 2 The torque about any fixed reference point o due to external forces acting on the system is defined as N ( o ) ( t ) = X r ( t )- o F ( e ) ( t ) , (2) where F ( e ) ( t ) is the force acting on the -th particle by objects external to the system. Corollary 1 If the systems total linear momentum is zero, its angular momentum does not depend on the choice of the reference point. Corollary 2 If the vector sum of all external forces acting on the system is zero, the torque does not depend on the choice of the reference point. Theorem 1 The rate of change of a systems angular momentum about any fixed refer- ence point is equal to the total external torque exerted on the system about the same reference point: d dt L ( o ) ( t ) = N ( o ) ( t ) . (3) To prove this, we note that d dt L ( o ) ( t ) = X r ( t ) m r ( t ) + X [ r ( t )- o ] m r ( t ) . Because the vector product of any two parallel vectors is zero, d dt L ( o ) ( t ) = X [ r ( t )- o ] m r ( t ) = X [ r ( t )- o ] F ( e ) ( t ) + X f ( t ) = N ( o ) ( t ) + X [ r ( t )- o ] f ( t ) 1 Since and and dummy indices, their names can be interchanged, so the last term is equal to...
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