Two spherical planets A and B have the same m | vedclass

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(A) Given: $\rho_A : \rho_B = 8 : 1$ and $M_A = M_B = M$.Since density $\rho = \frac{M}{V} = \frac{M}{\frac{4}{3}\pi R^3}$,we have $\rho \propto \frac

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$4:1$

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(A) Given: $\rho_A : \rho_B = 8 : 1$ and $M_A = M_B = M$.Since density $\rho = \frac{M}{V} = \frac{M}{\frac{4}{3}\pi R^3}$,we have $\rho \propto \frac{1}{R^3}$.Therefore,$\frac{\rho_A}{\rho_B} = \left(\frac{R_B}{R_A}\right)^3 = 8$.Taking the cube root,$\frac{R_B}{R_A} = 2$,which means $\frac{R_A}{R_B} = \frac{1}{2}$.The acceleration due to gravity is given by $g = \frac{GM}{R^2}$.Thus,the ratio $\frac{g_A}{g_B} = \frac{GM/R_A^2}{GM/R_B^2} = \left(\frac{R_B}{R_A}\right)^2$.Substituting the value,$\frac{g_A}{g_B} = (2)^2 = 4$.So,the ratio is $4:1$.

Two spherical planets $A$ and $B$ have the same mass,but their densities are in a ratio $8:1$. For these planets,the ratio of acceleration due to gravity at the surface of $A$ to its value at the surface of $B$ is:

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