A smooth bead $B$ of mass $0.3{\text{ }}kg$ is threaded on a light inextensible string of length $0.9{\text{ }}m$. One end of the string is attached to a fixed point $A$, and the other end of the string is attached to a fixed point $C$ which is vertically below $A$. The tension in the string is $T{\text{ }}N$, and the bead rotates with angular speed $\omega {\text{ }}rad{\text{ }}{s^{ - 1}}$ in a horizontal circle about the vertical axis through $A$ and $C$.

a) Given that $B$ moves in a circle with centre $C$ and radius $0.2{\text{ }}m$, calculate $\omega $, and hence find the kinetic energy of $B$.

b) Given instead that angle $ABC = {90^ \circ }$, and that $AB$ makes an angle ${\tan ^{ - 1}}\left( {\frac{1}{2}} \right)$  with the vertical, calculate $T$ and $\omega $.