Question

Consider a rectangular plate of dimension $a\times b$ If this plate is considered to be made up of four rectangles of dimension $\frac{a}{2}\times \frac{b}{2}$ and we now remove one out of four of the four rectangles, find the positions where the centre of mass of the remaining system will lie?

Answer 1

Given,

Initial mass of rectangular plate is $m$

After cutting one forth mass, remaining mass is $\frac{m}{2}+\frac{m}{4}=\frac{3m}{4}$

From figure,@

$OB=\frac{b}{4}andOA=\sqrt{{\left(\frac{a}{4}\right)}^2+{\left(\frac{b}{4}\right)}^2}=\frac{\sqrt{a^2+b^2}}{2}$

Center of mass,

$X_{com}=\frac{\frac{m}{4}\times \frac{\sqrt{a^2+b^2}}{2}+\frac{m}{2}\times \frac{b}{4}}{\frac{m}{4}+\frac{m}{2}}=\frac{\sqrt{a^2+b^2}+b}{6}$

Center of mass from center O is $\frac{\sqrt{a^2+b^2}+b}{6}$