Shown in the figure above is circle \(O\) with radius 6 inches. Chord \(CD\) is drawn perpendicular to radius \(AO\) so that its midpoint is 3 inches from the center of the circle. From point \(A\), any chord \(AB\) is drawn intersecting \(CD\) at point \(M\). Let \(v\) be equal to the product \((AB)(AM)\), as chord \(AB\) is made to rotate in the circle about the fixed point \(A\). Find \(v\).