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A circular ho...

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 :

\frac{a R^{2}}{R^{2} - r^{2}}

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\frac{a r^{2}}{R^{2} - r^{2}}

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\frac{a (R^{2} - r^{2})}{r^{2}}

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\frac{a (R^{2} - r^{2})}{R^{2}}

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Solution

The correct option is B $\frac{a r^{2}}{R^{2} - r^{2}}$
Suppose the $n e w$ center of mass is at a distance $X_{r e m}$ from the $c e n t e r$ of $b i g g e r$ disc and the center of mass of the
hole will be at a distance $X_{r m d} = a$ from the center of the bigger disk.

Now area of bigger disk is $π R^{2}$ and that of hole is $π a^{2}$

denoting the area of $r e m o v e d$ part as $A_{r m d} = π r^{2}$

and the area of $r e m a i n i n g$ part as $A_{r e m} = π (R^{2} - r^{2})$

and using the formula $X_{r e m} A_{r e m} = X_{r m d} A_{r m d}$

we get $X_{r e m} = a \frac{r^{2}}{R^{2} - r^{2}}$

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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 :

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 :