A block of mass $m$ placed on a smooth horizontal surface is connected by two elastic strings ab and cd of equal length $l$ , Young’s Modulus $Y$ and $2 Y$ , area of cross section $A$ and $\frac{A}{2}$ respectively, and a spring of force constant C. Both the strings and spring are in unstretched condition and if the time period of small oscillations of the block along the horizontal surface is $2 π [\sqrt{\frac{m}{x_{1} C}} + \sqrt{\frac{m l}{x_{2} Y A + x_{3} C l}}]$ . Find $x_{1} + x_{3} - x_{2}$ .

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Detailed Solution

$K_{1} = \frac{Y A}{l}, K_{2} = \frac{Y A}{l}, K_{3} = C$
Equivalent $K_{s t r i n g} = \frac{2 Y A}{l}$
$t_{1} = \frac{T_{1}}{2} = π \sqrt{\frac{m}{(\frac{2 Y A}{l} + C)}}, t_{2} = \frac{T_{2}}{2} = π \sqrt{\frac{m}{C}}$
Total time period of oscillation = $t_{1} + t_{2} = 2 π [\sqrt{\frac{m}{4 C}} + \sqrt{\frac{m l}{8 Y A + 4 C l}}]$