The mass density inside a solid sphere of rad | vedclass

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(C) The mass $M(r)$ enclosed in a sphere of radius $r$ is given by:$M(r) = \int_0^r \rho(r') \cdot 4\pi r'^2 dr' = \int_0^r \rho_0 \left(\frac{r'}{R}\

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$3$

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(C) The mass $M(r)$ enclosed in a sphere of radius $r$ is given by:$M(r) = \int_0^r \rho(r') \cdot 4\pi r'^2 dr' = \int_0^r \rho_0 \left(\frac{r'}{R}\right)^\beta \cdot 4\pi r'^2 dr' = \frac{4\pi \rho_0}{R^\beta} \int_0^r r'^{\beta+2} dr' = \frac{4\pi \rho_0}{R^\beta} \cdot \frac{r^{\beta+3}}{\beta+3}$.For $r = \frac{R}{2}$,the mass enclosed is $M_1 = \frac{4\pi \rho_0}{R^\beta} \cdot \frac{(R/2)^{\beta+3}}{\beta+3} = \frac{4\pi \rho_0 R^3}{\beta+3} \cdot \frac{1}{2^{\beta+3}}$.The gravitational field at $r = \frac{R}{2}$ is $E_1 = \frac{G M_1}{(R/2)^2} = \frac{G}{(R/2)^2} \cdot \frac{4\pi \rho_0 R^3}{(\beta+3) 2^{\beta+3}} = \frac{16 G \pi \rho_0 R}{(\beta+3) 2^{\beta+3}}$.For $r = 2R$,the total mass of the sphere is $M_2 = \frac{4\pi \rho_0}{R^\beta} \cdot \frac{R^{\beta+3}}{\beta+3} = \frac{4\pi \rho_0 R^3}{\beta+3}$.The gravitational field at $r = 2R$ is $E_2 = \frac{G M_2}{(2R)^2} = \frac{G}{4R^2} \cdot \frac{4\pi \rho_0 R^3}{\beta+3} = \frac{G \pi \rho_0 R}{\beta+3}$.Given $\frac{E_2}{E_1} = 4$:$\frac{\frac{G \pi \rho_0 R}{\beta+3}}{\frac{16 G \pi \rho_0 R}{(\beta+3) 2^{\beta+3}}} = 4 \Rightarrow \frac{2^{\beta+3}}{16} = 4 \Rightarrow 2^{\beta+3} = 64 = 2^6$.Therefore,$\beta + 3 = 6$,which gives $\beta = 3$.

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