A particle of mass $^{'} m^{'}$ is attached to three identical springs $A, B$ and $C$ each of force constant $^{'} k^{'}$ as shown in figure. If the particle of mass $^{'} m^{'}$ is pushed sightly against the spring $^{'} A^{'}$ and released. Find the period of oscillation.

Question

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Answer 1

$F_{n e t} = F_{A} + F_{B} cos 4 5^{0} + F_{C} cos 4 5^{0}$
$= k y + 2 k y^{'} cos 4 5^{0} = k y + 2 k (y cos 45^{0}) cos 4 5^{0} = 2 k y$

$F_{n e t} = k^{'} y \Rightarrow k^{'} y = 2 k y \Rightarrow k^{'} = 2 k$
$T = 2 π \frac{m}{k ^{'}} = 2 π \frac{m}{2 k}$
we get, period of oscillation is $2 π \frac{m}{2 k} .$

Answer 2

F_{n e t} = F_{A} + F_{B} cos 4 5^{0} + F_{C} cos 4 5^{0}

= k y + 2 k y^{'} cos 4 5^{0} = k y + 2 k (y cos 45^{0}) cos 4 5^{0} = 2 k y

F_{n e t} = k^{'} y \Rightarrow k^{'} y = 2 k y \Rightarrow k^{'} = 2 k

T = 2 π \frac{m}{k ^{'}} = 2 π \frac{m}{2 k}

we get, period of oscillation is

2 π \frac{m}{2 k} .