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Question

A circular hole of radius $$r$$ is made in a disk of radius $$R$$ and of uniform thickness at a distance $$a$$ from the centre of the disk. The distance of the new centre of mass from the original centre of mass is : 

40044_6a91dffcb4c040a48b8ee2d9b467a99a.png


A
aR2R2r2
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B
ar2R2r2
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C
a(R2r2)r2
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D
a(R2r2)R2
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Solution

The correct option is B $$\displaystyle \frac{ar^{2}}{R^{2}-r^{2}}$$
Suppose the $$new$$ center of mass is at a distance $$X_{rem} $$ from the $$center$$ of $$bigger $$ disc and the center of mass of the 
hole will be at a distance $$X_{rmd}=a$$ from the center  of the bigger disk.

Now area of bigger disk is $$\pi R^2$$ and that of hole is $$\pi a^2$$

denoting the area of $$removed$$ part as $$A_{rmd}=\pi r^2$$

and the area of $$remaining$$ part as $$A_{rem}=\pi (R^2-r^2)$$

and using the formula  $$X_{rem}A_{rem}=X_{rmd}A_{rmd}$$

we get $$X_{rem}=a\dfrac{r^2}{R^2-r^2}$$

Physics

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