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In the diagram shown, no friction at any contact surface. Initially, the spring has no deformation. What will be the maximum deformation in the spring? Consider all the strings to be sufficiency large. Consider the spring constant to be $K$.
$4F / 3K$
$8F / 3K$
$F / 3K$
none
Solution
Work done by pseudoframe in $C -$frame is zero. $W_{1 F}=F x_{1 C}$
$W_{2 F}=2 F x_{2 C}$
$0-\frac{1}{2} k x_{\mathrm{max}}^{2}+F X_{1 C}+2 F x_{2 C}=0…….ii$
$\frac{1}{2} k_{\max }^{2}=F x_{1 C}+2 F x_{2 C}$
$x_{\max }=x_{1 C}+x_{2 C}$
From the concept of centre of mass
$M x_{1 C}=2 M x_{2 C}$
$X_{1 C}=2 x_{2 C}$
$\therefore x_{1 C}+2 X_{2 C}=x_{\max }$
$\therefore 3 x_{2 C}=x_{\max }$
$x_{2 C}=\frac{x_{\max }}{3}$ and $x_{1 C}=\frac{2 x_{\max }}{3}$ Puttng the values in eqn ii
$\frac{1}{2} k x_{\max }^{2}=F\left(\frac{2 x_{\max }}{3}\right)+2 F\left(\frac{x_{\max }}{3}\right)$
$\frac{1}{2} k x_{\max }=\frac{2 F}{3}+\frac{2 F}{3}$
$x_{\max }=\frac{8 F}{3 K}$
