(I) $
\begin{aligned}
& \mathrm{V}_{\text {net }}=\sqrt{2} \omega \mathrm{R}=\sqrt{2} \\
& \mathrm{I} \rightarrow \mathrm{~S}
\end{aligned}
$
(II) $\mathrm{v}_{\mathrm{A}}(\mathrm{t}=0.13)=\frac{5 \pi}{\sqrt{2}} \cos 45 \hat{\mathrm{i}}+\left[\frac{5 \pi}{\sqrt{2}} \times \frac{1}{\sqrt{2}}-\mathrm{g} \times 0.1\right] \hat{\mathrm{j}}$
$
\begin{aligned}
& =\frac{5 \pi}{\sqrt{2}} \hat{i}+\left(\frac{5 \pi}{2}-1\right) \hat{j} \\
& v_B(t=0.1 \sec )=\frac{-5 \pi}{2} \hat{i}+\left(\frac{5 \pi}{2}\right) \hat{j}
\end{aligned}
$
After $\mathrm{t}=0.1$, relative velocities should not change.
$
\begin{aligned}
& \mathrm{V}_{\mathrm{rel}}(\mathrm{t}=0.1 \mathrm{sec})=|5 \pi \hat{\mathrm{i}}-\hat{\mathrm{j}}| \\
& =\sqrt{25 \pi^2+1}
\end{aligned}
$
$\mathrm{II} \rightarrow \mathrm{T}$
(III) $\mathrm{x}=\mathrm{x}_{\mathrm{A}}-\mathrm{x}_{\mathrm{B}}$
$
=x_0 \sin t-x_0 \sin \left(t+\frac{\pi}{2}\right)
$
$\begin{aligned} & =\sqrt{2} x_0 \sin \left(t-\frac{\pi}{4}\right) \\ & V_{\text {rel }}=\frac{d x}{d t}=\sqrt{2} x_0 \cos \left(t-\frac{\pi}{4}\right) \\ & =\sqrt{2} \cos \left(\frac{\pi}{3}-\frac{\pi}{4}\right) \\ & =\sqrt{2} \times \frac{\sqrt{3}+1}{2 \sqrt{2}}=\frac{\sqrt{3}+1}{2} \\ & \text { III } \rightarrow P \\ & \text { (IV) } V_{\text {rel }}=\sqrt{3^2+1^2}=\sqrt{10} \\ & \text { IV } \rightarrow R\end{aligned}$
