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Question:-
A sphere of radius 'r' is removed from a sphere of radius 'R' as shown in the figure. The distance between the centres of the spheres is 'd'. Mass is unformly distributed throughout the large whole sphere. Find the position of centre of mass of the new system.
Solution:-
Since the sphere is uniform, density of both large and small spheres are same.
[tex]\longrightarrow\sf{\dfrac{M}{V}=\dfrac{m}{v}}[/tex]
Here,
But the volume of the spheres are given by,
Then,
[tex]\longrightarrow\sf{\dfrac{M}{\frac{4}{3}\,\pi R^3}=\dfrac{m}{\frac{4}{3}\,\pi r^3}}[/tex]
[tex]\longrightarrow\sf{m=\dfrac{M}{\frac{4}{3}\,\pi R^3}\cdot\frac{4}{3}\,\pi r^3}[/tex]
[tex]\longrightarrow\sf{m=M\cdot \dfrac{r^3}{R^3}\quad\dots(1)}[/tex]
Initially, let centre of mass of the large sphere be at zero position.
Then centre of mass of the small sphere will be at 'd' units right from this zero position.
As small sphere is 'removed' from large sphere, position of new centre of mass of the system will be,
[tex]\longrightarrow\sf{\overline{x}_{new}=\dfrac{M\overline{X}-m\overline{x}}{M-m}}[/tex]
[tex]\longrightarrow\sf{\overline{x}_{new}=\dfrac{M(0)-m(d)}{M-m}}[/tex]
[tex]\longrightarrow\sf{\overline{x}_{new}=-\dfrac{md}{M-m}}[/tex]
From (1),
[tex]\longrightarrow\sf{\overline{x}_{new}=-\dfrac{\left(M\cdot\frac{r^3}{R^3}\right)d}{M-\left(M\cdot\frac{r^3}{R^3}\right)}}[/tex]
[tex]\longrightarrow\sf{\overline{x}_{new}=-\dfrac{M\left(\frac{r^3d}{R^3}\right)}{M\left(1-\frac{r^3}{R^3}\right)}}[/tex]
Cancelling M,
[tex]\longrightarrow\sf{\overline{x}_{new}=-\dfrac{\frac{r^3d}{R^3}}{\left(1-\frac{r^3}{R^3}\right)}}[/tex]
[tex]\longrightarrow\sf{\overline{x}_{new}=\dfrac{\frac{r^3d}{R^3}}{\left(\frac{r^3}{R^3}-1\right)}}[/tex]
[tex]\longrightarrow\sf{\overline{x}_{new}=\dfrac{\frac{r^3d}{R^3}}{\left(\frac{r^3-R^3}{R^3}\right)}}[/tex]
[tex]\longrightarrow\sf{\underline{\underline{\overline{x}_{new}=\dfrac{r^3d}{r^3-R^3}}}}[/tex]
This is the position of new centre of mass. This is a negative value since [tex]\sf{R > r\implies r^3-R^3 < 0}[/tex] depicting that the new centre of mass is at left of centre of large sphere, which was considered as zero position.