Question

The ratio of acceleration due to gravity at a depth $h$ below the surface of earth and at a height $h$ above the surface for $h<<R$

• A. constants only when $h<<R$
• B. increases linearly with $h$
• C. increases parabolically with $h$
• D. decreases

Answer 1

Answer: (B)

increases linearly with $h$

Above the surface, $g_{above}^{\prime }=\frac{GM}{(R+h{)}^2}$ (at depth h)

Inside the earth, $g_{below}^{\prime }=\frac{GM}{R^2}(1-\frac{h}{R})$

$\frac{g_{below}^{\prime }}{g_{above}^{\prime }}=\frac{\frac{1}{R^2}(1-\frac{h}{R})}{\frac{1}{(R+h{)}^2}}=\frac{1-\frac{h}{R}}{\frac{2}{{(1+\frac{h}{R})}^2}}$

But for $\frac{h}{R}<<<1$

$\frac{g_{below}^{\prime }}{g_{above}^{\prime }}$=$(1-\frac{h}{R}){(1+\frac{h}{R})}^2$

$=(1-\frac{h}{R})(1+\frac{2h}{R}+{\left(\frac{h}{R}\right)}^2)=1+\frac{2h}{R}+{\left(\frac{h}{R}\right)}^2-\frac{h}{R}-2{\left(\frac{h}{R}\right)}^2-{\left(\frac{h}{R}\right)}^3=1+\frac{h}{R}$

It increases linearly with ${}^{\prime }{h}^{\prime }$