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

The depth $d$ at which the value of acceleration due to gravity becomes $\frac{1}{n}$ times the value at the surface, is $[R=$ radius of the earth $]$
(a) $\frac{R}{n}$
(b) $R\left(\frac{n-1}{n}\right)$
(c) $\frac{R}{n^{2}}$
(d) $R\left(\frac{n}{n+1}\right)$

Key Concepts

Relationship Between Depth and Radial Distance

When analyzing effects within a spherical body, it is useful to relate the depth below the surface to the corresponding radial distance from the center. The gravitational acceleration is expressed in terms of this radial distance, and the problem can be framed by equating the ratio of this distance to the total radius with the given fraction of the surface gravity.

Uniform Density Assumption

This concept assumes that the mass of the sphere (such as the Earth) is uniformly distributed throughout its volume. Under this assumption, the gravitational field inside the sphere can be derived more straightforwardly, leading to a linear relation between the gravitational acceleration and the distance from the center.

Shell Theorem

The shell theorem states that a spherically symmetric shell of mass exerts no net gravitational force on a particle located inside it. Consequently, when calculating the gravitational acceleration at a point within a uniform sphere, only the mass enclosed within the radius up to that point contributes to the gravitational pull.

Linear Variation of Gravitational Acceleration Inside a Sphere

For a body with uniform density, the gravitational acceleration inside varies linearly with the radial distance from the center. This means that if you move from the center to the surface, the acceleration increases in a directly proportional manner, which allows for simple scaling of gravity at different depths.

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