A square plate of edge d and a circular disc of diameter d are placed touching each other at the midpoint of an edge of the plate as shown in figure. Locate the centre of mass of the combination, assuming same mass per unit area for the two plates. 
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Let m be the mass per unit area.
$\therefore$ Mass of the square plate = $M_1 = d^2m$ Mass of the circular disc $=\text{M}_2=\frac{\pi\text{d}^2}{4}\text{m}$ Let the centre of the circular disc be the origin of the system.
$\therefore$ Position of centre of mass
$=\Bigg(\frac{\text{d}^2\text{md}+\pi\Big(\frac{\text{d}^2}{4}\Big)\text{m}\times0}{\text{d}^2\text{m}+\pi\Big(\frac{\text{d}^2}{4}\Big)\text{m}},\frac{0+0}{\text{M}_1+\text{M}_2}\Bigg)$
$=\Bigg(\frac{\text{d}^3\text{m}}{\text{d}^2\text{m}\big(1+\frac{\pi}{4}\big)},0\Bigg)$
$=\Big(\frac{\text{4d}}{\pi+4},0\Big)$ The new centre of mass is $\Big(\frac{\text{4d}}{\pi+4}\Big)$ right of the centre of circular disc.
$\therefore$ Mass of the square plate = $M_1 = d^2m$ Mass of the circular disc $=\text{M}_2=\frac{\pi\text{d}^2}{4}\text{m}$ Let the centre of the circular disc be the origin of the system.
$\therefore$ Position of centre of mass
$=\Bigg(\frac{\text{d}^2\text{md}+\pi\Big(\frac{\text{d}^2}{4}\Big)\text{m}\times0}{\text{d}^2\text{m}+\pi\Big(\frac{\text{d}^2}{4}\Big)\text{m}},\frac{0+0}{\text{M}_1+\text{M}_2}\Bigg)$
$=\Bigg(\frac{\text{d}^3\text{m}}{\text{d}^2\text{m}\big(1+\frac{\pi}{4}\big)},0\Bigg)$
$=\Big(\frac{\text{4d}}{\pi+4},0\Big)$ The new centre of mass is $\Big(\frac{\text{4d}}{\pi+4}\Big)$ right of the centre of circular disc.
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