Physics

Physics

Problem

For small displacements from the surface, the expressions for gg at height hh and at depth dd may be approximated. Note: The explicit instruction to use a binomial expansion has been removed. a. (i) Starting from the exact expression for gg at altitude, obtain a simplified relation showing that the fractional change in gg is approximately −2hR-\frac{2h}{R} when h≪Rh \ll R. b. (ii) Repeat the procedure for depth dd to show that the fractional change in gg is approximately −dR-\frac{d}{R} when d≪Rd \ll R. c. (iii) Using results of (i) and (ii), evaluate the ratio fractional change per metre of depthfractional change per metre of altitude\frac{\text{fractional change per metre of depth}}{\text{fractional change per metre of altitude}} near the surface and interpret its physical meaning.
Intuitive explanation

The key insight is that gravity changes predictably as you move away from Earth's center, and for small distances compared to Earth's radius, you can use a linear approximation. Whether you go up (height hh) or down (depth dd), the gravitational field weakens or strengthens based on how the distance from Earth's center changes—and that change is tiny relative to RR. Treating gg as a linear function of position near the surface reveals a beautiful symmetry: the rate of change is the same whether you're slightly above or slightly below.

Intuitive explanation

The key insight is that gravity changes predictably as you move away from Earth's center, and for small distances compared to Earth's radius, you can use a linear approximation. Whether you go up (height hh) or down (depth dd), the gravitational field weakens or strengthens based on how the distance from Earth's center changes—and that change is tiny relative to RR. Treating gg as a linear function of position near the surface reveals a beautiful symmetry: the rate of change is the same whether you're slightly above or slightly below.

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