Figure 1 shows a hemispherical bowl of internal radius $$\(a\)$$ fixed with its open plane face uppermost and horizontal. The lowest point of the bowl is $$\(A\)$$. A light inextensible string of length $$\(a \sqrt{3}\)$$ has one end fixed to the point $$\(B\)$$, where $$\(B\)$$ is vertically above $$\(A\)$$ and $$\(A B=2 a\)$$. A particle, $$\(P\)$$, of mass $$\(m\)$$ is attached to the other end of the string. The particle moves in a horizontal circle on the smooth inner surface of the bowl with constant angular speed $$\(\omega\)$$. The string remains taut and the particle remains in contact with the bowl throughout the motion. (a) Find, in terms of $$\(m, a, \omega\)$$ and $$\(g\)$$, the tension in the string. (7) (b) Show that $$\(\omega \geqslant \sqrt{\frac{2 g}{3 a}}\)$$ (4)

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