A light spring of natural length \(a\) and modulus of elasticity \(kmg\) is attached to a fixed point \(O\) on a smooth plane inclined at angle \(\theta\) to the horizontal, where \(\sin\theta=\dfrac34\). A particle of mass \(m\) is attached to the lower end of the spring and is held at point \(A\), where \(OA=2a\) and \(OA\) is along a line of greatest slope of the plane.

The particle is released from rest. It is moving with speed \(V\) when it passes through point \(B\), where \(OB=\dfrac32a\). Its speed is \(\dfrac12V\) when it passes through point \(C\), where \(OC=\dfrac34a\).

Find \(k\).