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(B) When the particle of mass $m$ at $O$ is pushed by a small displacement $y$ in the direction of spring $A$,spring $A$ is compressed by $y$. Springs

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$2\pi \sqrt{\frac{m}{2k}}$

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(B) When the particle of mass $m$ at $O$ is pushed by a small displacement $y$ in the direction of spring $A$,spring $A$ is compressed by $y$. Springs $B$ and $C$ are stretched by an amount $y' = y \cos 45^\circ$.The total restoring force on the mass $m$ along the direction $OA$ is:$F_{net} = F_A + F_B \cos 45^\circ + F_C \cos 45^\circ$$F_{net} = ky + (ky') \cos 45^\circ + (ky') \cos 45^\circ$$F_{net} = ky + 2k(y \cos 45^\circ) \cos 45^\circ$$F_{net} = ky + 2ky \cos^2 45^\circ = ky + 2ky(1/2) = ky + ky = 2ky$Comparing this with $F_{net} = k_{eff} y$,we get $k_{eff} = 2k$.The time period of oscillation is given by $T = 2\pi \sqrt{\frac{m}{k_{eff}}}$.Substituting $k_{eff} = 2k$,we get $T = 2\pi \sqrt{\frac{m}{2k}}$.

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

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