Answer
$$
s_C=78.0 \mathrm{~mm}
$$
Work Step by Step
$$
v_C=\omega_B(0.1)=0.03 \omega_A
$$
If $\omega_B=6 \mathrm{rad} / \mathrm{s}$ then
$$
\begin{aligned}
& \omega_A=20 \mathrm{rad} / \mathrm{s} \\
& v_C=0.6 \mathrm{~m} / \mathrm{s} \\
& T_1+V_1=T_2+V_2
\end{aligned}
$$
$$
T_1+V_1=T_2+V_2
$$
$$
[0+0+0]+[0]=\frac{1}{2}\left[\frac{1}{2}(3)(0.03)^2\right](20)^2+\frac{1}{2}\left[\frac{1}{2}(10)(0.1)^2\right](6)^2+\frac{1}{2}(2)(0.6)^2-2(9.81) s_c
$$
$$
s_C=78.0 \mathrm{~mm}
$$
