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A uniform metal disc of radius $R$ is taken and out of it a disc of diameter $\displaystyle\frac{R}{2}$ is cut-off from the end. Find the center of mass of remaining part?
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![here is the answer for the above question. 100% correct][1]

Density is uniform, so we can use geometrical method.

Area of disc, A1 =πR² Area of cutoff disc, A2 = (π/4)(R/2)² = πR²/16

Xcom = (A1×0-A2X2)/(A1-A2) = ((πR²/16)×(3R/4))/((πR²)- ( πR²/16)) =

-3R/15×4 = -R/20

Hence, center of mass is at R/20 from the center.

Area of disc, $\mathrm{A}_{1}=\pi \mathrm{R}^{2}$
Area of cutoff disc, $\mathrm{A}_{2}=\frac{\pi}{4}\left(\frac{\mathrm{R}}{2}\right)^{2}=\frac{\pi \mathrm{R}^{2}}{16}$ $$\begin{array}{l} x_{\text {com }}=\frac{A_{1} \times 0-A_{2} x_{2}}{A_{1}-A_{2}}=\frac{-\frac{\pi R^{2}}{16} \times \frac{3 R}{4}}{\pi R^{2}-\frac{\pi R^{2}}{16}}=\frac{-3 R}{15 \times 4}= \\ -\frac{R}{20} \end{array}$$ Hence, center of mass is at $\frac{\mathrm{R}}{20}$ from the center.