A spherical cavity of radius r is curved out | vedclass

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Question

(d)

Centre of mass of solid sphere is at origin (centre of sphere).

Now, centre of mass of sphere with cavity is calculated as follows.

$x_{

Vedclass · Accepted Answer

$\frac{r^3}{R^2+R r+r^2}$

Vedclass · Answer

(d)

Centre of mass of solid sphere is at origin (centre of sphere).

Now, centre of mass of sphere with cavity is calculated as follows.

$x_{ CM }=\frac{m_1 x_1-m_2 x_2}{m_1-m_2}$

$=\frac{\left(\frac{4}{3} \pi R^3 \cdot \rho\right)(0)-\left(\frac{4}{3} \pi r^3 \cdot \rho\right)(R-r)}{\frac{4}{3} \pi R^3 \rho-\frac{4}{3} \pi r^3 \rho}$

where, $\rho=$ density of material of sphere.

$\Rightarrow \quad x_{ CM }=-\frac{r^3(R-r)}{\left(R^3-r^3\right)}$

$=\frac{-r^3(R-r)}{(R-r)\left(R^2+R r+r^2\right)}$

$=\frac{-r^3}{R^2+R r+r^2}$

So, distance between two mass centres is

$d=\frac{r^3}{R^2+R r+r^2}$

A spherical cavity of radius $r$ is curved out of a uniform solid sphere of radius $R$ as shown in the figure below. The distance of the centre of mass of the resulting body from that of the solid sphere is given by

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