Question

show that the rate of change of angular momentum of system is equal to sum of external torque acting on the system ​

Answer 1

Answer:

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Explanation:

Theorem:

The rate of change of the total angular momentum of a system of particles is equal to the sum of the external torques on the system.

Thus:

L=∑iri×pi(3.11.1)

∴L˙=∑ir˙i×p˙i(3.11.2)

But the first term is zero, because  r˙  and  pi  are parallel.

Also

r˙i=Fi+∑Fij(3.11.3)

L˙i=∑iri×(ri+∑jFij)=∑iri×Fi+∑iri×∑jFii  

∴∑iri×Fi+∑iri×∑jFii  

But  ∑i∑jFij=0  by Newton’s third law of motion, and so  ∑i∑jri×Fij=0 .

Also  ∑iri×Fi=τ  , and so we arrive at

L˙=τ(3.11.4)

which was to be demonstrated.