A particle of mass m is attached to three identical springs A, B and C each of force constant k a 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'=ycos45°.  So that the total restoring force on the mass m along OA.Fnet=FA+FBcos45°+FCcos45°=ky+2ky'cos45°=ky+2k(ycos45°)cos45°=2kyAlso Fnet=k'y⇒k'y=2ky⇒k'=2kT=2πmk'=2πm2k