Physics

Problem

Near the Earth’s surface, g changes with both altitude h and depth d. a. (i) Starting from the exact expressions

g_h = \dfrac{GM}{(R+h)^2}

and

g_d = g\left(1 - \dfrac{d}{R}\right)

, obtain the first-order rates of change

\dfrac{dg_h}{dh}

and

\dfrac{dg_d}{dd}

evaluated at

h = 0

and

d = 0

respectively. b. (ii) Hence find the ratio of the magnitudes of these two rates and interpret its physical meaning. c. (iii) On the same set of axes draw neat, labelled qualitative graphs of g versus the fractional quantities

h/R

(for altitude) and

d/R

(for depth) up to 0.3. Mark the linear segments and the region where deviation from linearity begins for each curve.

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