Question

Find the magnitude and direction of the force acting on the particle of mass $m$ during its motion in the plane xy according to the law $x=a\sin \omega t$, $y=b\cos \omega t$, where $a$, $b$, and $\omega$ are constants.

Answer 1

Obviously the radius vector describing the position of the article relative to the origin of coordinate is
$\overrightarrow{r}=x\overrightarrow{i}+y\overrightarrow{j}=a\sin \omega t\overrightarrow{i}+b\cos \omega t\overrightarrow{j}$
Differentiating twice w.r.t time
$\overrightarrow{w}=\frac{d^2\overrightarrow{r}}{d{t}^2}=-{\omega }^2(a\sin \omega t\overrightarrow{i}+b\cos \omega t\overrightarrow{j})=-{\omega }^2\overrightarrow{r}$ (1)
Thus $\overrightarrow{F}=m\overrightarrow{w}=-m{\omega }^{2}r$