A particle of mass 'm' is attached to three identical springs `A,B` and `C` each of force constant 'K' as shown in figure. If the particle of mass 'm' is pushed slightly against the spring 'A' and released the period of oscillations is

Options

A. `2pisqrt((2m)/(k))`
B. `2pisqrt((m)/(2k))`
C. `2pisqrt((m)/(k))`
D. `2pisqrt((m)/(3k))`

Answer

B

Explanation

When the particle of mass m at O is pushed by x, the spring A will be compressed by x while springs B and C will be stretched by `x.=x cos 45^(@)`, so that the total restoring force on the mass m along AO
`=F_(A)+F_(B)cos45^(@)+F_(C)cos45^(@)`
`=-kx-kx.cos45^(@)-kx.cos45^(@)=-(kx+2kx.cos45^(@))`
`=-(kx+2k(xcos45^(@))cos45^(@)) ("as "x.=xcos45^(@))`
`=-2kx=-kx("as "cos45^(@)=(1)/(sqrt(2)))`
where `k.=2x`
`therefore T=2pisqrt((m)/(k.))=2pisqrt((m)/(2k))`