PX155 - E5 - moments

introduction

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

m o m e n t = F \times d_{p e r p}

- take moments about $P$ : $r_{1} W_{1} = r_{2} W_{2}$ for balance

torque

define torque:

\vec \tau = \vec r \times \vec F$$with dimensions $Nm$ - generalisation of turning moment to 3D ![Pasted image 20231104143249.png](/img/user/pics/Pasted%20image%2020231104143249.png) - perpendicular distance of line of action of $\vec F$ to point $O$ is $r\sin\theta$ - from the *right-hand rule*: $\vec\tau$ out of the page generating an anti-clockwise rotation of particle - like a moment, $\vec \tau$ must always be defined with respect to a point (of reference), not necessarily the centre of rotation ### derivation - what does a torque do? - fromnewton's second law: $\vec F = m \frac{d\vec v}{dt}$ $$\vec r \times \vec F = m (\vec r\times\frac{d\vec v}{dt})

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

\vec{r} \times \vec{F} = m \frac{d}{d t} (\vec{r} \times \vec{v}) = \frac{d}{d t} (\vec{r} \times \vec{p})

- looks a lot like newton's second law , so, define $\vec{L} = \vec{r} \times \vec{p}$ ( $k g m^{2} s^{- 1}$ ), 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 $\vec{τ}$ and $\vec{L}$ must be around some point)
note: a particle moving in a straight line will have a non-zero $\vec{L}$ along any point not along that straight line