Variation with Depth

Using Shell Theorem: only mass below depth d contributes to g.

Mass below depth d: M' = M(R−d)^{3}/$R^{3}$

g at depth d: g'(d) = GM'/（R−d)^{2} = g(1 − d/R)

Key points:

• Linear decrease (not inverse-square)
• g = 0 at Earth's centre (d = R)
• Depth formula is gentler than altitude formula
• Finding depth where g = g/n: d = R(1 − 1/n)

Variation with Latitude λ

Earth's rotation creates a centrifugal pseudo-force at latitude λ:

g_eff(λ) = g − Rω^{2}c$os^{2}$λ

where ω = $7.27 \times 10^{-5}$ rad $s^{-1}$ (Earth's angular velocity).

Key values:

• At pole (λ = 90°): cos λ = 0, g_eff = g (maximum)
• At equator (λ = 0°): cos λ = 1, g_eff = g − Rω^{2} (minimum)
• Difference: Rω^{2} ≈ 0.034 m $s^{-2}$ ≈ 0.3% of g

In practice, Earth's oblate shape adds another ~0.2% difference between pole and equator.

NEET Quick Comparison Rule

At equal distance x from surface:

• Altitude: g' = g/(1 + x/R)^{2} < g(1 − x/R) = depth value
• Altitude always gives lower g than depth at equal distance