circular motion

Force constant of the string k = mg/e
where m is the mass of the weight, and g is the acceleration due to gravity.

Let T be the tension in the string
T.cos(theta) = mg --------------------- (1)where (theta) is the angle the string makes with the verticalBut T = k.x = (mg/e)x, where x is the extension of the stringHence, (mg/e)x.cos(theta) = mgcos(theta) = e/x ----------------- (2)
For the circular motion, T.sin(theta) = m.r.w^2 -------------- (3).where r is the radius of the circle
But r = (a+x).sin(theta), Then (3) becomes, T.sin(theta) = m.[(a+x).sin(theta)].w^2i.e. (mg/e).x.sin(theta) = m[(a+x).sin(theta)].w^2gx/e = (a+x).w^2solve for x gives x = aew^2/(g-ew^2) substitute the value of x into (2)cos(theta) = (g-ew^2)/aw^2 ------------------- (4)i.e. (theta) = arc-cos[(g-ew^2)/aw^2] From (4), (aw^2).cos(theta) = g-ew^2i.e. w^2 = g/[a.cos(theta) + e]The values of (theta) must lie between 0 and 90 degrees, i.e. 1 > cos(theta) > 0Thus, g/(a+e) < w^2 < g/e