Consider a sphere of mass $M$ and radius $R$ centered at origin. The density of material of the sphere is $\rho=A r^\alpha$, where $r$ is the radial distance, $\alpha$ and $A$ are constants. If the moment of inertia of the sphere about the axis passing through centre is $\frac{6}{7} M R^2$, then the value of $\alpha$ is

Given, density of sphere, $p=A r^\alpha$ (where, $r=$ radial distance and $A$ and $\alpha$ are constants) Consider an elemental spherical shell of radius $r$ and thickness $d r$
Mass of elemental spherical shell, $$ \begin{aligned} & d m=\text { Volume } \times \text { Density } \\ & d m=\left(4 \pi r^2\right) d r \cdot A r^\alpha=4 \pi A r^{2+\alpha} d r \end{aligned} $$ Mass of entire solid sphere, $$ \begin{aligned} & M=4 \pi A \int_0^R r^{2+\alpha} d r \\ & M=4 \pi A\left[\frac{r^{3+\alpha}}{3+\alpha}\right]_0^R=\frac{4 \pi A}{3+\alpha} \cdot R^{3+\alpha} \end{aligned} $$ Now, moment of inertia of elemental spherical shell is $$ d I=\frac{2}{3}(d m) \cdot r^2=\frac{2}{3}\left(4 \pi A r^{2+\alpha} d r\right) r^2 $$ Moment of inertia of entire solid sphere, $$ I=\int_0^R d I=\frac{2}{3} 4 \pi A \int_0^R r^{4+\alpha} \cdot d r $$ $$ \begin{aligned} & I=\frac{2}{3} 4 \pi A\left[\frac{r^{5+\alpha}}{5+\alpha}\right]_0^R \Rightarrow I=\frac{2}{3} 4 \pi A\left[\frac{R^{5+\alpha}}{5+\alpha}\right] \\ & I=\frac{2}{3}\left(\frac{4 \pi A}{3+\alpha} \cdot R^{3+\alpha}\right) \cdot \frac{R^2(3+\alpha)}{5+\alpha} \end{aligned} $$ From Eq. (ii), we get $$ I=\frac{2}{3} M R^2 \cdot\left(\frac{3+\alpha}{5+\alpha}\right) $$ It is given in the question, $$ I=\frac{6}{7} M R^2 $$ On comparing Eqs (iii) and (iv), we get $$ \begin{aligned} \frac{2}{3} M R^2 \cdot\left(\frac{3+\alpha}{5+\alpha}\right) & =\frac{6}{7} M R^2 \Rightarrow \frac{3+\alpha}{5+\alpha}=\frac{9}{7} \\ 21+7 \alpha & =45+9 \alpha \\ \alpha \quad \alpha & =-12 \end{aligned} $$