Answer
$$
v_A=1.40 \mathrm{~m} / \mathrm{s}
$$
Work Step by Step
$$
\begin{aligned}
& T_1+V_1=T_2+V_2 \\
& {[0+0+0]+[0+0]=\frac{1}{2}\left[3(0.045)^2\right] \omega^2+\frac{1}{2}(2)(0.03 \omega)^2+\frac{1}{2}(2)(0.1 \omega)^2-2(9.81) s_A+2(9.81) s_B} \\
& \theta=\frac{s_B}{0.03}=\frac{s_A}{0.1} \\
& s_B=0.3 s_A \\
& \text { Set } s_A=0.2 \mathrm{~m}, s_B=0.06 \mathrm{~m}
\end{aligned}
$$
Substituting and solving yields,
$$
\begin{aligned}
& \omega=14.04 \mathrm{rad} / \mathrm{s} \\
& v_A=0.1(14.04)=1.40 \mathrm{~m} / \mathrm{s}
\end{aligned}
$$
