Question

The acceleration due to gravity at a point A at certain height from surface of the earth is $\frac{g}{4}$ where 'g' is acceleration due to gravity on the surface of the earth. At a point B which lies vertically at certain height above point A, the acceleration due to gravity is $\frac{g}{9}$. The distance between points A and B is (R - Radius of the earth)}

• A. $5R$
• B. $3R$
• C. $R$
• D. $2R$

Hint:

Remember: \[ g_h=g\left(\frac{R}{R+h}\right)^2 \] At height \(R\), gravity becomes \(\frac{g}{4}\).

Answer 1

The Correct Option is C

Concept: Acceleration due to gravity at a height \(h\) above the earth's surface is given by \[ g_h=g\left(\frac{R}{R+h}\right)^2 \] where \(R\) is the radius of the earth.

Step 1: Determine the height of point A.
Given, \[ \frac{g}{4}=g\left(\frac{R}{R+h_A}\right)^2 \] Cancelling \(g\), \[ \left(\frac{R}{R+h_A}\right)^2=\frac14 \] \[ \frac{R}{R+h_A}=\frac12 \] \[ R+h_A=2R \] \[ h_A=R \]

Step 2: Determine the height of point B.
Given, \[ \frac{g}{9}=g\left(\frac{R}{R+h_B}\right)^2 \] \[ \left(\frac{R}{R+h_B}\right)^2=\frac19 \] \[ R+h_B=3R \] \[ h_B=2R \]

Step 3: Calculate the distance between A and B.
\[ AB=h_B-h_A \] \[ AB=2R-R \] \[ AB=R \] Hence, \[ \boxed{AB=R} \]