Question

If the change in the value of $g$ at a height $h$ above the surface of the earth is the same as at a depth $x$ below it when both $x$ and $h$ are much smaller than the radius of the earth, then

• A. $x=h$
• B. $x=2h$
• C. $x=\frac{h}{2}$
• D. $x=\frac{h}{3}$

Answer 1

Answer: (B)

$x=2h$

Since value of g at height h and at depth x is equal, we have

$g\left[\frac{R^2}{(R+h{)}^2}\right]=g\left[\frac{r-x}{R}\right]$

or, $\frac{R^2}{(R+h{)}^2}=\frac{R-x}{R}$

or, $R^3=(R-x)(R+h{)}^2$

$=(R-x)(R^2+2Rh+h^2)$

or, $2R^{2}h+R{h}^2-x{R}^2-2Rhx-x{h}^2=0$

or, $R(2Rh-xR-2hx)=0$

Considering h and x very small, we have

$x=2h$