Question

A ball of mass $m$ falls down without initial velocity from a height $h$ over the Earth's surface. Find the increment of the ball's angular momentum vector picked up during the time of falling (relative to the point $O$ of the reference frame moving translationally in a horizontal direction with a velocity $V$). The ball starts fallingfrom the point $O$. The air drag is to be neglected.

Answer 1

During the free fall time $t=\tau =\sqrt{\frac{2h}{g}}$, the reference point $O$ moves in horizontal direction (say towards right) by the distance $V\tau$. In the translating frame as $\overrightarrow{M}\left(O\right)=0$, so
$\Delta \overrightarrow{M}={\overrightarrow{M}}_f=\overrightarrow{r}$
$=(-V\tau \overrightarrow{i}+h\overrightarrow{j})\times m[g\tau \overrightarrow{j}-V\overrightarrow{i}]$
$=-mVg{\tau }^2\overrightarrow{h}+mVh(+\overrightarrow{k})$
$=-mVg\left(\frac{2h}{g}\right)\overrightarrow{k}+mVh(+\overrightarrow{k})=-mVh\overrightarrow{k}$
Hence $|\Delta \overrightarrow{M}|=mVh$