A bead $B$ of mass $m\text{ kg}$ travels with constant speed round a horizontal circle on a smooth fixed wire. The wire is a circle with centre $O$ and radius $0.4\text{ m}$. One end of a light elastic string, whose natural length is $0.4\text{ m}$ and modulus of elasticity is $42m\text{ N}$, is fastened to $B$. The other end of the string is fixed at point $A$, which is $0.3\text{ m}$ vertically above $O$ (see diagram).
(i)[3]
Show that the wire exerts a vertical contact-force component of $3.7m\text{ N}$ upwards on the bead.
(ii)[2]
Given that the contact force has no horizontal component, find the angular speed of $B$.
(iii)[3]
If instead the horizontal component of the contact force is of magnitude $2m\text{ N}$, find the two possible speeds of $B$.
(iv)[3]
Find the speed of bead $B$.
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