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

Let the particle of mass m at O be pushed by distance y in the direction of A. The spring A will be compressed by y, while sprinmgs B and each will be stretched by `y^(')(=ycos45^(@))` . Total restoring force on the mass m along AO is
`-F=F_(a)+F_(b)cos45^(@)+F_(c)45^(@)=ky+2(ky^('))cos45^(@)`
`=ky+2k(ycos45^(@))cos45^(@)=2ky`
or `F=-2ky` ...(i)
As, `Fpropy` and F is directed towards the equilibrium position O, hence mass m will execute linear S.H.M.
Comparing (i) with `F=-Ky`, we note that spring factor, `K=2k`
and inertia factor `=` mass of particle `=`m
Time period, `T=2pisqrt((i n ertial fact o r)/(spri ng fact o r))=2pisqrt((m)/(2k))`