A spring of force constant k and negligible mass has an unst

A spring of force constant k and negligible mass has an unstretched length d. A ball
of mass m is attached to one end of the spring, and the assembly is made to rotate in a circle about the
other end of the spring, which is attached to a fixed point. If the spring stretch is x, what is the period
of rotation T in terms of given constants?

Solution

Centrifugal force F_c = m²r

Tension in spring F_s = kx

Since the ball is rotating in equalibrium, F_c=F_s

²=kx/mr

r=d+x

²=(kx)/md(1+x/d),

= SQRT[(kx)/md(1+x/d)]

Period = 2/ = 2/SQRT[(kx)/md(1+x/d)] Seconds