question_answer

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

A)            \[2\pi \sqrt{\frac{2m}{k}}\]

B)            \[2\pi \sqrt{\frac{m}{2k}}\]

C)            \[2\pi \sqrt{\frac{m}{k}}\]

D)            \[2\pi \sqrt{\frac{m}{3k}}\]

Correct Answer: B

Solution :

                   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. \[{{F}_{net}}={{F}_{A}}+{{F}_{B}}\cos 45{}^\circ +{{F}_{C}}\cos 45{}^\circ \] \[=ky+2k{y}'\cos 45{}^\circ \]\[=ky+2k(y\cos 45{}^\circ )\cos 45{}^\circ \]\[=2ky\] Also \[{{F}_{net}}={k}'y\] Þ \[{k}'y=2ky\]Þ \[{k}'=2k\] \[T=2\pi \sqrt{\frac{m}{{{k}'}}}=2\pi \sqrt{\frac{m}{2k}}\]