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AngularMomentum

# AngularMomentum - Angular Momenta and Torques Shina Tan...

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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 α [ r α ( t ) - o ] × m α ˙ r α ( t ) , (1) where [ r α ( t ) - o ] is the position vector of the α -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 α r α ( t ) - o × F ( e ) α ( t ) , (2) where F ( e ) α ( t ) is the force acting on the α -th particle by objects external to the system. Corollary 1– If the system’s 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 system’s 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 α ˙ r α ( t ) × m α ˙ r α ( t ) + X α [ r α ( t ) - o ] × m α ¨ r α ( t ) . Because the vector product of any two parallel vectors is zero, d dt L ( o ) ( t ) = X α [ r α ( t ) - o ] × m α ¨ r α ( t ) = X α [ r α ( t ) - o ] × F ( e ) α ( t ) + X β f αβ ( t ) = N ( o ) ( t ) + X αβ [ r α ( t ) - o ] × f αβ ( t ) 1

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Since α and β and dummy indices, their names can be interchanged, so the last term is equal to βα [ r β ( t ) - o ] × f βα ( t ). According to Newton’s Third Law, f βα ( t ) = - f αβ ( t ). So the last term is also equal to
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AngularMomentum - Angular Momenta and Torques Shina Tan...

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