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Torque and Angular Momentum

Finally we come to the formally trickiest transformation of the SECOND LAW, the one involving the vector product (or ``cross product'') of   $\mbox{\boldmath$\vec{F}$\unboldmath }$  with the distance   $\mbox{\boldmath$\vec{r}$\unboldmath }$  away from some origin<sup>11.15</sup> ``O.'' Here goes:

$$\mbox{\boldmath $\vec{r}$\unboldmath } \; \times \; \left[ . . . . . . $\unboldmath } \times \mbox{\boldmath $\vec{F}$\unboldmath }$$

Now, the distributive law for derivatives applies to cross products, so

$${d \over dt} \, [ \mbox{\boldmath $\vec{r}$\unboldmath } \tim . . . . . . } \times {d\mbox{\boldmath $\vec{p}$\unboldmath } \over dt}$$

but

$${d\mbox{\boldmath $\vec{r}$\unboldmath } \over dt} \; \equiv . . . . . . h } \; \equiv \; m \, \mbox{\boldmath $\vec{v}$\unboldmath }$$

$$\hbox{\rm so} \qquad {d\mbox{\boldmath $\vec{r}$\unboldmath . . . . . . h } \times \mbox{\boldmath $\vec{v}$\unboldmath }) \; = \; 0$$

because the cross product of any vector with itself is zero.^11.16 Therefore

$${d \over dt} \, [ \mbox{\boldmath $\vec{r}$\unboldmath } \t . . . . . . unboldmath } \times \mbox{\boldmath $\vec{F}$\unboldmath } .$$

If we define two new entities,

$$\mbox{\boldmath$\vec{r}$\unboldmath } \times \mbox{\boldmat . . . . . . math } \; \equiv \; \mbox{\boldmath$\vec{L}$\unboldmath }_O,$$ (11.18)

$$\hbox{\rm the {\sl Angular Momentum\/} about } O$$

and

$$\mbox{\boldmath$\vec{r}$\unboldmath } \times \mbox{\boldmat . . . . . . th } \; \equiv \; \mbox{\boldmath$\vec{\tau}$\unboldmath }_O,$$ (11.19)

$$\hbox{\rm the {\sl Torque\/} generated by } \mbox{\boldmath $\vec{F}$\unboldmath } \hbox{\rm ~about } O \, ,$$

then we can write the above result in the form

$${d \mbox{\boldmath$\vec{L}$\unboldmath }_O \over dt} = \mbox{\boldmath$\vec{\tau}$\unboldmath }_O$$ (11.20)

This equation looks remarkably similar to the SECOND LAW. In fact, it is the rotational analogue of the SECOND LAW. It says that

``The rate of change of the angular momentum of a body about the origin  O  is equal to the torque generated by forces acting about  O.''

So what? Well, if we choose the origin cleverly this ``new'' Law gives us some very nice generalizations. Consider for instance an example which occurs very often in physics: the central force.