5.1 Moments

When dealing with problems involving rotation, we have mechanical quantities which are analogous to ones used to solve linear problems. We have seen a few of these before: angular velocity $\omega =v_{\theta }/r$, angular acceleration $\alpha =a_{\theta }/r$ (for circular motion), and moment of inertia $I=m{r}^2$. As we can see, the definitions of these quantities all involve the analogous linear quantity and the distance from the origin $r$. Quantities that involve the product of a linear quantity with the radius are called moments. Moments we have seen already are the center of mass, which is the first moment of mass (sum of $mr$) normalised by the total mass, and the moment of inertia, which is the second moment of mass (mass multiplied by $r^2$) In this chapter we will introduce two more moments: torque — the moment of force — and angular momentum — the moment of momentum. When vector quantities like force and momentum are involved, moments are defined using the cross product.

Note that because the cross product is antisymmetric, the order of $𝒓$ and $𝒑$ matters! The motivation for introducing angular momentum will become clear in the next section.