One end of a light elastic string of natural length $0.4{\text{ }}m$ and modulus of elasticity $20{\text{ }}N$ is attached to a particle $P$ of mass $0.8{\text{ }}kg$. The other end of the string is attached to a fixed point $O$ at the top of a smooth plane inclined at ${30^ \circ }$ to the horizontal. The particle rests in equilibrium on the plane.
a) Calculate the extension of the string.
$P$ is projected from its equilibrium position up the plane along a line of greatest slope. In the subsequent motion $P$ just reaches $O$, and later just reaches the foot of the plane. Calculate
b) the speed of projection of $P$,
c) the length of the line of greatest slope of the plane.
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