A ball of a mass m is connected to two rubber bands of length L, each under tension T as shown in figure . The ball is displaced by a small distance y perpendicular to the length of the rubber band. Assuming the tension does not change , shown that (a) the restoring force is `- (2T//L)y` and (b) the system exhibits simple harmonic motion with an angular frequency `omega = sqrt(2T//mL)`s.

Explanation

a. If we displace the block upward the net tension towards mean position will proved the restoration force on the block.
`sum bar F = - 2T sin theta`, where `theta = tan^(-1) (y//L)`

Since for a small displacement.
`sin theta = tan theta = (y)/(L)`
and the resultant is
`sum bar F = (- (2 Ty)/(L))`
b. Since there is a restoring force that is proportional to the position , it cause the system to move with simple harmonic motion like a block-spring system . Thus, we can think of
`omega = sqrt((k)/(m)) = sqer((2 T)/(mL))`
Not enything with a stable equilirium position will exercute simple harmonic motion when disturb and released, but a lot of things do when the disturbance is small.