Torque/Proofs

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This mathematics article is devoted entirely to providing mathematical proofs and support for claims and statements made in the article Torque. This article is currently an experimental vehicle to see how well we can provide proofs and details for a math article without cluttering up the main article itself. See Wikipedia:WikiProject Mathematics/Proofs for some current discussion. This article is "experimental" in the sense that it is a test of one way we may be able to incorporate more detailed proofs in Wikipedia.

A proof that torque is equal to the time-derivative of angular momentum can be stated as follows:

The definition of angular momentum for a single particle is:

$\mathbf{L} = \mathbf{r} \times \mathbf{p}$

where "×" indicates the vector cross product. The time-derivative of this is:

$\frac{d\mathbf{L}}{dt} = \mathbf{r} \times \frac{d\mathbf{p}}{dt} + \frac{d\mathbf{r}}{dt} \times \mathbf{p}$

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 = mv, we can see that:

$\frac{d\mathbf{L}}{dt} = \mathbf{r} \times m \frac{d\mathbf{v}}{dt} + \mathbf{v} \times m\mathbf{v}$

But the cross product of any vector with itself is zero, so the second term vanishes. Hence with the definition of force F = ma, (Newton's 2nd law) we obtain:

$\frac{d\mathbf{L}}{dt} = \mathbf{r} \times \mathbf{F}$

And by definition, torque τ = r×F. Note that there is a hidden assumption that mass is constant — this is quite valid in non-relativistic mechanics. Also, total (summed) forces and torques have been used — it perhaps would have been more rigorous to write: