The radii of two planets are respectively R1 and R2 and their densities are respectively ρ1 and

step 1 In the question it is given that the radius of two planets respectively R1 and R2 and the densit

The radii of two planets are respectively R1 and R2 and...

The radii of two planets are respectively $\mathrm{R}_{1}$ and $\mathrm{R}_{2}$ and their densities are respectively $\rho_{1}$ and $\rho_{2}$ the ratio of the accelerations due to gravity at their surface is (A) $g_{1}: g_{2}=\left(\rho_{1} / R_{1}^{2}\right) \cdot\left(\rho_{2} / R_{2}^{2}\right)$ (B) $\mathrm{g}_{1}: \mathrm{g}_{2}=\mathrm{R}_{1} \mathrm{R}_{2}: \rho_{1} \rho_{2}$ (C) $g_{1}: g_{2}=R_{1} \rho_{2} \cdot R_{2} p_{1}$ (D) $g_{1}: g_{2}=R_{1} \rho_{1}: R_{2} \rho_{2}$

The radii of two planets are respectively $\mathrm{R}_{1}$ and $\mathrm{R}_{2}$ and their densities are respectively $\rho_{1}$ and $\rho_{2}$ the ratio of the accelerations due to gravity at their surface is
(A) $g_{1}: g_{2}=\left(\rho_{1} / R_{1}^{2}\right) \cdot\left(\rho_{2} / R_{2}^{2}\right)$
(B) $\mathrm{g}_{1}: \mathrm{g}_{2}=\mathrm{R}_{1} \mathrm{R}_{2}: \rho_{1} \rho_{2}$
(C) $g_{1}: g_{2}=R_{1} \rho_{2} \cdot R_{2} p_{1}$
(D) $g_{1}: g_{2}=R_{1} \rho_{1}: R_{2} \rho_{2}$

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Key Concepts

Newton's Law of Universal Gravitation

This fundamental law determines that every mass attracts every other mass via a gravitational force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centers. It provides the basis for calculating forces and accelerations due to gravity in systems such as planetary bodies.

Relationship Between Mass, Volume, and Density

For any object, mass is determined by its density multiplied by its volume. In the case of spherical objects like planets, the volume is proportional to the cube of the radius. This connection between density, radius, and mass is crucial when deriving formulas for gravitational acceleration on the surface of a sphere.

Surface Gravity of a Spherical Body

Surface gravity is the acceleration due to gravity at the surface of a spherical body and can be derived from Newton's law. By substituting the expression for the mass of a sphere (in terms of density and volume) into the gravitational formula, one finds that surface gravity is directly proportional to the product of the radius and the density of the sphere. This relationship is key to comparing gravitational accelerations between different planetary bodies.

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