[tex]\begin{gathered} For\text{ smaller gear} \\ r=\text{ radius} \\ v_{Ts}=0.2\text{ m/s} \\ \text{For larger gear} \\ r_l=31\operatorname{cm}=0.31m \\ v_{Ts}=v_{Tl} \\ v_T=wr \\ \text{hence} \\ w_sr=w_lr_l \\ \text{Solving }w_l \\ w_l=\frac{w_sr}{r_l} \\ \text{but} \\ v_{Ts}=w_sr \\ \text{Solving }w_s \\ w_s=\frac{v_{Ts}}{r}=\frac{0.2\text{ m/s}}{r} \\ \text{Now} \\ w_l=\frac{0.2\text{ m/s}}{r}\cdot\frac{_{}r}{0.31m_{}} \\ w_l=0.65\text{ rad/s}\cdot\frac{r}{r} \\ \text{The angular velocity of larger gear is }0.65\text{ rad/s}\cdot\frac{r}{r} \\ \\ \text{Question b} \\ 2w_s \\ 2w_sr=w_lr_l \\ \text{Solving w}_l \\ w_l=\frac{2w_sr}{_{}r_l} \\ \text{If the angular velocity of the smaller gear is doubled the angular velocity} \\ of\text{ larger gear is doubled too} \end{gathered}[/tex]
