The depth of at which the value of acceleration due to...

The depth of at which the value of acceleration due to gravity becomes $1 / \mathrm{n}$ the time the value of at the surface is $(\mathrm{R}=$ radius of earth $)$ (A) $\mathrm{R} / \mathrm{n}$ (B) $R[(\mathrm{n}-1) / \mathrm{n}]$ (C) $\left(\mathrm{R} / \mathrm{n}^{2}\right)$ (D) $\mathrm{R}[\mathrm{n} /(\mathrm{n}+1)]$

The depth of at which the value of acceleration due to gravity becomes $1 / \mathrm{n}$ the time the value of at the surface is $(\mathrm{R}=$ radius of earth $)$
(A) $\mathrm{R} / \mathrm{n}$
(B) $R[(\mathrm{n}-1) / \mathrm{n}]$
(C) $\left(\mathrm{R} / \mathrm{n}^{2}\right)$
(D) $\mathrm{R}[\mathrm{n} /(\mathrm{n}+1)]$

Key Concepts

Uniform Density Earth Model

This concept assumes that the Earth has a constant density throughout its volume. Under this assumption, the gravitational force inside the Earth varies in a linear fashion with the distance from the center. This simplification is crucial because it allows us to derive simple relationships for the gravitational acceleration at different depths inside the Earth.

Gravitational Field Variation Inside a Sphere

According to the shell theorem, a spherically symmetric body affects external objects as if all its mass were concentrated at its center, and inside the sphere, only the mass enclosed by the radius at which you are calculating the field contributes to the gravitational force. For a body with uniform density, this means that the gravitational acceleration decreases linearly with the distance from the center, being zero at the center and reaching its maximum at the surface.

Relationship between Subsurface Depth and Radial Distance

When determining the depth beneath the Earth's surface where the gravitational acceleration is reduced to a fraction of its surface value, it is important to relate the radial distance from the center of the Earth to the depth below the surface. The gravitational acceleration at a distance r from the center is proportional to r/R, so setting this ratio equal to a given fraction allows us to solve for r and then determine the corresponding depth, which is found by subtracting r from the Earth's radius.

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