The definition of angular momentum for a single particle is:
where "×" indicates the vector cross product. The timederivative of this is:
This result can easily be proven by splitting the vectors into components and applying the product rule. Now using the definitions of velocity v = dr/dt, acceleration a = dv/dt and linear momentum p = ma, we can see that:
But the cross product of any vector with itself is zero, so the second term vanishes. Hence with the definition of force F = ma, we obtain:
And by definition, torque τ = r×F. Note that there is a hidden assumption that mass is constant — this is quite valid in nonrelativistic mechanics. Also, total (summed) forces and torques have been used — it perhaps would have been more rigorous to write:
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