PX155 - E5 - moments

introduction

• familiar eg:
  • archimedes' lever rule: moments must balance if the lever doesn't rotate

- take moments about $P$: $r_1{W}_1=r_2{W}_2$ for balance

torque

• define torque:

- consider $\frac{d}{dt}(\overrightarrow{r}\times \overrightarrow{v})=\frac{d\overrightarrow{r}}{dt}\times \overrightarrow{v}+\overrightarrow{r}\times \frac{d\overrightarrow{v}}{dt}$
- $\frac{d\overrightarrow{r}}{dt}=\overrightarrow{v},\therefore \frac{d\overrightarrow{r}}{dt}\times \overrightarrow{v}=0$
$$\frac{d}{dt} (\vec r \times \vec v) = \vec r \times \frac{d\vec v}{dt}$$
- so,

- looks a lot like newton's second law , so, define $\overrightarrow{L}=\overrightarrow{r}\times \overrightarrow{p}$ ($kg{m}^2{s}^{-1}$), which is the angular momentum (moment of momentum)
$$\vec \tau = \vec r \times \vec F = \frac{d}{dt}\vec L$$

• torque is the rate of change of angular momentum (both $\overrightarrow{\tau }$ and $\overrightarrow{L}$ must be around some point)
• note: a particle moving in a straight line will have a non-zero $\overrightarrow{L}$ along any point not along that straight line