A uniform rectangular thin sheet A B C D of mass M has length a and breadth b , as shown in the figure. If…

A uniform rectangular thin sheet $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ of mass $\mathrm{M}\mathrm{ }$ has length $\mathrm{a}$ and breadth $\mathrm{b}$, as shown in the figure. If the shaded portion $\mathrm{H}\mathrm{B}\mathrm{G}\mathrm{O}$ is cut-off, the coordinates of the centre of mass of the remaining portion will be:

1. $\frac{5\mathrm{a}}{12},\frac{5\mathrm{b}}{12}$
2. $\frac{5\mathrm{a}}{3},\frac{5\mathrm{b}}{3}$
3. $\frac{2\mathrm{a}}{3},\frac{2\mathrm{b}}{3}$
4. $\frac{3\mathrm{a}}{4},\frac{3\mathrm{b}}{4}$

Solution

If the total mass of object mass 'M' then the mass removed part will be $\frac{\mathrm{M}}{4}$ .

${\mathrm{X}}_{\mathrm{C}\mathrm{M}}=\frac{\mathrm{M}·\frac{\mathrm{a}}{2}-\frac{\mathrm{M}}{4}\left(\frac{3\mathrm{a}}{4}\right)}{\mathrm{M}-\frac{\mathrm{M}}{4}}=\frac{\frac{\mathrm{M}\mathrm{a}}{2}-\frac{3\mathrm{M}\mathrm{b}}{16}}{\frac{3\mathrm{M}}{4}}=\frac{5}{12}\mathrm{a}$
${\mathrm{Y}}_{\mathrm{C}\mathrm{M}}=\frac{\mathrm{M}·\frac{\mathrm{b}}{2}-\frac{\mathrm{M}}{4}\left(\frac{3\mathrm{b}}{4}\right)}{\mathrm{M}-\frac{\mathrm{M}}{4}}=\frac{\frac{\mathrm{M}\mathrm{b}}{2}-\frac{3\mathrm{M}\mathrm{a}}{16}}{\frac{3\mathrm{M}}{4}}=\frac{5}{12}\mathrm{b}$

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