Physics - Gravitation Question with Solution | TestHub

Given: ${\rho }_1:{\rho }_2=1:8$ and $m_1=m_2$ We know that, $\rho =\frac{\mathrm{m}}{\mathrm{V}}$ $$\begin{array}{rl}& \Rightarrow {\rho }_1{V}_1={\rho }_2{V}_2\\ & \Rightarrow {\rho }_1\frac{4}{3}\pi R_1^3={\rho }_2\frac{4}{3}\pi R_2^3\\ & \Rightarrow \left(\frac{R_1}{R_2}\right)=\frac{{\rho }_2}{{\rho }_1}=\frac{8}{1}\\ & \Rightarrow \left(\frac{R_1}{R_2}\right)=\frac{2}{1}\end{array}$$ As we know that the acceleration due to gravity, $g=\frac{GM}{R^2}$ $$\begin{array}{rl}& g=\frac{G}{R^2}\times \rho \times \frac{4}{3}\pi R^3=\frac{4}{3}G\pi \rho R\\ & \Rightarrow \frac{g_1}{g_2}=\left(\frac{{\rho }_1}{{\rho }_2}\right)\left(\frac{R_1}{R_2}\right)\\ & \therefore \frac{g_1}{g_2}=\frac{1}{8}\times 2=\frac{1}{4}\text{ New soln }\end{array}$$