Problem

$A$ and $B$ are two fixed points at a distance $22a$ apart, with $B$ vertically below $A$. A light elastic string of natural length $4a$ and modulus of elasticity $5mg$ has one end attached to $A$ and the other end attached to a particle $P$ of mass $km$. Another light elastic string of natural length $8a$ and modulus of elasticity $6mg$ has one end attached to $B$ and the other end attached to $P$. Particle $P$ is vertically above $B$.

(a) Show that, when the system is in equilibrium,

$$BP=\frac{57a-2ak}{4}.$$

The particle $P$ is pulled vertically upwards so that $BP=18a$, and is then released from rest. In its subsequent motion, $P$ first comes to instantaneous rest at the point where $BP=8a$.

(b) Find the value of $k$.