Question

A uniform T-shaped arm of weight W, pinned about a horizontal point $C$, is supported by a vertical spring of stiffness $K$, The extension of the spring is

• A. $\frac{3W}{4K}$
• B. $\frac{4W}{3K}$
• C. $\frac{3K}{4W}$
• D. $\frac{4K}{3W}$

Answer 1

The correct option is B $\frac{4W}{3K}$


Center of gavity of T-shaped is at $\overline{x}$ distance from $C,$ calculated as
$\overline{x}=\frac{A_1{\overline{x}}_1+A_2{\overline{x}}_2}{A_1+A_2}$
$=\frac{(\frac{L}{2}+\frac{L}{2})\times 2L\times 2L\times L}{L+2L}=\frac{4}{3}L$


Taking moment about $C$
$\Rightarrow W\times \frac{4}{3}L=F\times L$
$\Rightarrow F=Kx=\frac{4}{3}W$
$\Rightarrow x=\frac{4}{3}\frac{W}{K}$