A particle of mass $m$ starts moving at $t$ = 0 due to a force $F=F_0$ $\sin \omega t$ where $F_0$ and $\omega$ are constant. Then correct statement is/are

$F=F_0 \sin \omega t=\frac{mdv}{dt}\Rightarrow {\int }_0^{v}dv={\int }_0^t\frac{F_0}{m}\sin \omega t dt$

$V=\frac{F_0}{m\omega }\left[1-\cos \omega t\right]$so at $t=\mathrm{\pi }/\mathrm{w}$

$V=\frac{2F_0}{m\omega } \therefore$(1) (a) is wrong. The max value of $V=\frac{F_0}{m\omega }$ when $\cos \omega t=0$. $\therefore$ (3) is correct.

Now $\frac{dS}{dt}=\frac{F_0}{m\omega }\left[1-\cos \omega t\right]$

${\int }_0^{s}dS={\int }_0^t\frac{F_0}{m\omega }\left[1-\cos \omega t\right]dt$

$S=\frac{F_0}{m{\omega }^2}\left[\omega t-\sin \omega t\right]$at $t=2\pi /\omega$.

$S=\frac{F_0\pi }{m{\omega }^2}$ $\therefore$ (2) is wrong.