A light elastic string has natural length $$\(a\)$$ and modulus of elasticity $$\(\frac{3}{4} m g\)$$. A particle $$\(P\)$$ of mass $$\(m\)$$ is attached to one end of the string. The other end of the string is attached to a fixed point $$\(A\)$$. Particle $$\(P\)$$ hangs freely in equilibrium at the point $$\(O\)$$, vertically below $$\(A\)$$. (a) Find the distance $$\(O A\)$$. (2) The particle $$\(P\)$$ is now pulled vertically down to a point $$\(B\)$$, where $$\(A B=3 a\)$$, and released from rest. (b) Show that, throughout the subsequent motion, $$\(P\)$$ performs only simple harmonic motion, justifying your answer. (6) The point $$\(C\)$$ is vertically below $$\(A\)$$, where $$\(A C=2 a\)$$. Find, in terms of $$\(a\)$$ and $$\(g\)$$, (c) the speed of $$\(P\)$$ at the instant that it passes through $$\(C\)$$, (3) (d) the time taken for $$\(P\)$$ to move directly from $$\(B\)$$ to $$\(C\)$$. (4)

Answer:

Knowledge points:

Solution: