Question

[Key Concept: A removed part is treated as a negative mass placed at its own centre of mass.]

$$x_{cm} = \frac{M_{1}x_{1} - M_{2}x_{2}}{M_{1} - M_{2}}$$

$$M = \sigma A = \sigma\pi r^{2}$$

$$M_{1} = \sigma\pi a^{2}\text{~at~}x_{1} = 0\& M_{2} = \sigma\pi b^{2}\text{~at~}x_{2} = c$$

$$\begin{matrix} & \ \\ & \ \\ x_{cm} & \ = \frac{\left( \sigma A_{1} \right)\left( x_{1} \right) - \left( \sigma A_{2} \right)\left( x_{2} \right)}{\sigma A_{1} - \sigma A_{2}} \\ & \ = \frac{\left( \pi a^{2} \right)(0) - \left( \pi b^{2} \right)(c)}{\pi a^{2} - \pi b^{2}} \\ & \ = - \left( \frac{b^{2}c}{a^{2} - b^{2}} \right) \end{matrix}$$

$y_{cm} = 0$, because of symmetry along x-axis.

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