A particle of mass m is moving along a trajectory given by $x=x_0+a\cos {\omega }_{1}t$ and $y=y_0+b\sin {\omega }_{2}t$. The torque, acting on the particle about the origin, at t=0 is:

$\overrightarrow{r}=\left(x_o+a\cos {\omega }_{1}t\right)i+\left(y_o+b\sin {\omega }_{2}t\right)j$;

 at t=0 $\overrightarrow{r}=\left(x_o+a\right)i+y_{o}j$

$\overrightarrow{a}=\frac{d^{2}x}{d{t}^2}\widehat{i}+\frac{d^{2}y}{d{t}^2}\widehat{j}$

$\overrightarrow{a}=(-a{\omega }_1^2\cos {\omega }_{1}t)\widehat{i}+(-b{\omega }_2^2\sin {\omega }_{2}t)j$; 

at t=0 $\overrightarrow{a}=-a{\omega }_1^2\widehat{i}$

$T=\overrightarrow{r}\times \overrightarrow{F}$$=m\left(\overrightarrow{r}\times \overrightarrow{a}\right)$$=m\left(y_o\left(-a{\omega }_1^2\right)-\widehat{k}\right)$

$=m{y}_{o}a{\omega }_1^2\widehat{k}$