The acceleration due to gravity is g at a point distance r...

The acceleration due to gravity is $g$ at a point distance $r$ from the center of earth $\mathrm{R}$. if $\mathrm{r}<\mathrm{R}$ then
(A) $g \propto r$
(B) $g \propto r^{2}$
(C) $g \propto \mathrm{r}^{-2}$
(D) $g \propto r^{-1}$

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Key Concepts

Uniform Density Assumption

The assumption of uniform density for the Earth is crucial when analyzing the gravitational field inside a spherical mass. Under this assumption, the mass enclosed within a radius r is directly proportional to r³, and when combining this with the gravitational force formula, it results in gravitational acceleration being proportional to r, thereby leading to a linear increase with distance from the center.

Gravitational Field Inside a Spherical Mass

Inside a uniformly dense spherical mass, the gravitational field increases linearly with distance from the center. This happens because only the mass enclosed within the sphere of radius r affects the gravitational force at that distance, leading to a proportional relationship between gravitational acceleration and r. This contrasts with the inverse square dependence observed outside the sphere.

Shell Theorem

The Shell Theorem in gravitation states that a spherically symmetric shell of mass exerts no net gravitational force on a particle located inside it. This theorem allows us to conclude that only the mass enclosed within a sphere of radius r (when r is less than the total radius R of the Earth) contributes to the gravitational force at that point, simplifying the analysis of gravitational fields inside spherical bodies.

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