Three springs A , B , and C , each having a force constant k , are...

Let the displacement of the particle be $x$ towards spring $C$.

The compression in spring $C$ is $x$. The restoring force exerted by spring $C$ is $F_C = kx$, directed opposite to the displacement.

The angle between the line of action of spring $C$ and those of springs $A$ and $B$ is 120^{}^{{}^{\circ}}.

The extension in springs $A$ and $B$ is the component of displacement along their respective lengths, which is x \cos(180^{}^{{}^{\circ}} - 120^{}^{{}^{\circ}}) = x \cos 60^{}^{{}^{\circ}} = \dfrac{x}{2}.

The restoring force exerted by each of springs $A$ and $B$ is $F_A = F_B = k \left(\dfrac{x}{2}\right), directed towards their respective fixed ends.<br><br>The net restoring force from springs$A$and$B$along the direction opposite to the displacement is:<br><br>$F_{A,B} = F_A \cos 60^{}^{{}^{\circ}} + F_B \cos 60^{}^{{}^{\circ}} = 2 \left(\dfrac{kx}{2}\right) \left(\dfrac{1}{2}\right) = \dfrac{kx}{2}

The total restoring force on the particle is:

$F_{net} = F_C + F_{A,B} = kx +$\dfrac{kx}{2} = \dfrac{3kx}{2}

The effective spring constant of the system is $k_{eff} =$\dfrac{3k}{2}.

The time period of oscillation is given by:

$T = 2\pi$\sqrt{\dfrac{m}{k_{eff}}} ==2\pi \sqrt{\dfrac{2m}{3k}}