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 and therefore the potential energy is proportional to &theta;², say that it is $$\tfrac{1}{2}k\theta^{2}$$. Now from the values of y and x above we have

For small oscillations we have x = 1 and tan &theta; = &theta;; and therefore

Hence the potential energy is

and the equation of motion of the particle becomes

Hence the period $$T_1$$ of oscillation is

In the second case — when the rod is parallel to the line of relative motion of S_1 and $$S_2$$ — the amount of twisting in the wire for a given position of the balls is the absolute value of $$(\pi/2)-\theta$$. The potential energy is $$\tfrac{1}{2}k\left[(\pi/2)-\theta\right]^{2}$$. We have

For small oscillations we have

Hence the potential energy is $$\tfrac{1}{2}kx^{2}$$ the period $$T_2$$ of oscillation is therefore

Equating the two periods of oscillation found above we have