The fixed points, $$\(A\)$$ and $$\(B\)$$, are a distance $$\(10 a\)$$ apart, with $$\(B\)$$ vertically above $$\(A\)$$. One end of a light elastic string, of natural length $$\(2 a\)$$ and modulus of elasticity $$\(2 m g\)$$, is attached to a particle $$\(P\)$$ of mass $$\(m\)$$ and the other end is attached to $$\(A\)$$. One end of another light elastic string, of natural length $$\(4 a\)$$ and modulus of elasticity $$\(6 m g\)$$, is attached to $$\(P\)$$ and the other end is attached to $$\(B\)$$. The particle $$\(P\)$$ rests in equilibrium at the point $$\(C\)$$, as shown in Figure 6. (a) Show that each string has an extension of $$\(2 a\)$$. (5) The particle $$\(P\)$$ is now pulled down vertically, so that it is a distance $$\(a\)$$ below $$\(C\)$$ and then released from rest. (b) Show that in the subsequent motion, $$\(P\)$$ performs simple harmonic motion. (4) (c) Find, in terms of $$\(a\)$$ and $$\(g\)$$, the speed of $$\(P\)$$ when it is a distance $$\(\frac{7}{2} a\)$$ above $$\(A\)$$. (4)

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