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 then the time period of oscillations is

When the particle of mass m at O is pushed by y in the direction of A The spring A will be compressed by y while spring B and C will be stretched by y′=ycos⁡45∘. So that the total restoring force on the mass m along OA. Fnet =FA+FBcos⁡45∘+FCcos⁡45∘=ky+2ky′cos⁡45∘=ky+2kycos⁡45∘cos⁡45∘=2kyAlso Fnet =k′y⇒k′y=2ky⇒k′=2kT=2πmk′=2πm2k