A particle of mass m is attached to four springs with spring constant k, k, 2k and 2k. as shown in the figure. Four springs are attached to the four corners of a square and a particle is placed at the center. If the particle is pushed slightly towards any sides of the square and released, the period of oscillation will be
 

Options

A. `2pi sqrt((m)/(3k))`
B. `2pi sqrt((m)/(3sqrt(2)k))`
C. `2pi sqrt((m)/(6k))`
D. `2pi sqrt((m)/(2k))`

Answer

B

Explanation

Along Y-axis, net force on the particle is
`F=-(k x sin 45^(@)+k x sin 45^(@)+2kx sin135^(@)+2kx sin135^(@))`
`=-(6)/(sqrt(2))kx`
`implies" "m(d^(2)x)/(dt^(2))=-3sqrt(2)kximplies(d^(2)x)/(dt^(2))+(3sqrt(2)kx)/(m)=0`
This is the equation of SHM.
Angular frequency, `omega^(2)=3sqrt(2)(k)/(m)`
`implies" "f=((1)/(2)pi)sqrt((3sqrt(2)k)/(m))impliesT=2pisqrt((m)/(3sqrt(2)k))`