Question

A block of mass m placed on a smooth horizontal surface is connected by two elastic strings ab and cd of equal length , Young’s Modulus Y & 2Y, area of cross section A & $\frac{A}{2}$ respectively and a spring of force constant C. Both the strings and spring are in unstretched condition, if the time period of small oscillation of the block along the horizontal surface is.
$2\pi [\sqrt{\frac{m}{x_{1}c}}+\sqrt{\frac{ml}{x_{2}YA+x_{3}cl}}].Find{x}_1+x_3-x_2$ ___

Answer 1

$K_1=\frac{YA}{1},K_2=\frac{YA}{1},K_3=C$
$Equivalent{K}_{string}=\frac{2YA}{1}$
$t_1=\frac{T_1}{2}=\pi \sqrt{\frac{m}{(\frac{2YA}{1}+C)}},t_2=\frac{T_2}{2}=\pi \sqrt{\frac{m}{c}}$
Total time period of oscillation
$=t_1+t_2=2\pi [\sqrt{\frac{m}{4C}}+\sqrt{\frac{ml}{8YA+4Cl}}]$