Question

The value of g at a certain height h above the free surface of earth is $\frac{x}{4}$, where x is the value of 'g' at the surface of earth. The height h is

• A.
  $R$
• B.
  $2R$
• C.
  $3R$
• D.
  $4R$

Answer 1

The correct option is A

$R$
Value of acceleration due to gravity on the surface of the earth, $x=g=\frac{GM}{R^2}$

Here, G = gravitational constant, M = mass of the earth and R = radius of the earth

Value of acceleration due to gravity at height $h,g_h=\frac{x}{4}=\frac{GM}{(R+h{)}^2}$

Taking ratio of $g_{h}andg,$ we get:

$\frac{g_h}{g}={\left(\frac{R}{R+h}\right)}^2\Rightarrow \frac{\frac{x}{4}}{x}={\left(\frac{R}{R+h}\right)}^2\Rightarrow {\left(\frac{R}{R+h}\right)}^2=\frac{1}{4}\Rightarrow \left(\frac{R}{R+h}\right)=\frac{1}{2}\Rightarrow h+R=2R\Rightarrow h=R$

Hence, the correct answer is option (a).