A particle of mass `m` is attached with three springs `A,B` and `C` of equal force constancts `k` as shown in figure. The particle is pushed slightly against the spring `C` and released. Find the time period of oscillation.
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Explanation

When the partical of mass `m` at `O` is pushed by `y` in the direction of `A` , spring A will be compressed by `y` while `B and C` will be stetched by `y = y cos 45^(@)` , so the total restoring force on mass `m` along`Ao`,
`RF = F_(A) + F_(B) cos 45^(@) + F_(C) cos 45^(@)`
`= ky + 2 (ky') cos 45^(@)`
`Ky + 2 k (y cos 45^(@)) cos 45^(@)`
`F = (2 k) y implies ma = - (2 k) y`
Hence `a = = ((2 k)/(m)) y` as compared with `a = - omega^(2) y`
We get `omega^(2) = (2 k)/(m)` which gives `T = 2 pi sqrt((m)/(2 k))`