Question

Find the relation between the gravitational field on the surface of two planets `A` and `B` of masses `m_(A), m_(B)` and radii `R_(A)` and `R_(B), `respectively if
a. they have equal mass.
b. they have equal (uniform)density.

Answer 1

Let `E_A` and `E_B` be the gravitational field intensities on the surface of planets A & B.
`E_A=(Gm_A)/R_A^2=(G4/3prR_A^3rho_A)/R_A^2=(4Gpi)/3rho_AR_A`
Similarly, `E_(B) = (Gm_(B))/(R_(B)^(2))=(4G)/(3)pirho_(B)R_(B)`
(i) for `m_(A) = m_(B) = (A_(A))/(E_(B)) = (R_(B)^(A))/(R_(A)^(2))`
(ii) For & `rho_(A) = rho_(B) = (E_(A))/(E_(B)) = (R_(A))/(R_(B))`