A particle of mass m is attached to three identical spr

SOLUTION

Sol Let the particle of mass m at O be pushed downward against spring A. Then the forces produced in springs B and C will be in the directions shown.             
If y is the instantaneous compression of spring A, then B and C will be stretched by $y\cos {45}^{\circ }$ each.
$\therefore$ Total restoring force on mass m along AO will be
                                                $F=F_A+F_B\cos {45}^{\circ }+F_C\cos {45}^{\circ }$
$=ky+\left(ky\cos {45}^{\circ }\right)\cos {45}^{\circ }+\left(ky\cos {45}^{\circ }\right)\cos {45}^{\circ }$
                          $=ky(1+2{\cos }^2{45}^{\circ })$
                          $=ky(1+2\frac{1}{2})=2ky$
$\Rightarrow$              
                                          $a=\frac{2ky}{m}\therefore T=2\pi \sqrt{\frac{m}{2k}}$
$\therefore T=2\pi \sqrt{\frac{m}{2k}}$