In the given figure, the block is attached with a system of three ideal springs A, B, C. The block is displaced by a small distance x from its
equilibrium position vertically downwards and released. T represents the time period of small vertical oscillations of the block. Then (pulleys
are ideal)

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$T = 2 π \sqrt{\frac{11 m}{2 k}}$

the deformation of the spring C is (1/11) times the displacement of the block

the deformation of the spring B is (4/11) times the displacement of the block.

the deformation of the spring A is (2/11) times the displacement of the block.

answer is A, B, C, D.

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Detailed Solution

Let elongation in spring A, B and C be x₁, x₂ and x₃ respectively. Considering spring forces and constraint relations
$\begin{array}{ll} x_{2} = 4 x_{3} \dots (i) \\ x_{2} = 2 x_{1} \dots (ii) \end{array}$
and $x_{1} + 2 x_{2} + x_{3} = x \dots (iii)$
$\Rightarrow x_{1} = (\frac{2}{11}) x; x_{2} = (\frac{4}{11}) x; x_{3} = (\frac{1}{11}) x$
Also, $F = 2 K (\frac{x}{11})$
$\Rightarrow T = 2 π \sqrt{\frac{11 m}{2 k}}$