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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
$2\pi \sqrt {\frac{{2m}}{k}} $
$2\pi \sqrt {\frac{m}{{2k}}} $
$2\pi \sqrt {\frac{m}{k}} $
$2\pi \sqrt {\frac{m}{{3k}}} $
Solution
(b) 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 $y' = y\cos 45^\circ .$ So that the total restoring force on the mass $m $ along $ OA.$
${F_{net}} = {F_A} + {F_B}\cos 45^\circ + {F_C}\cos 45^\circ $
$ = ky + 2ky'\cos 45^\circ $$ = ky + 2k(y\cos 45^\circ )\cos 45^\circ $$ = 2ky$
Also ${F_{net}} = k'y$ ==> $k'y = 2ky$==> $k' = 2k$
$T = 2\pi \sqrt {\frac{m}{{k'}}} = 2\pi \sqrt {\frac{m}{{2k}}} $
