Animated Solution for Physics - Gravitation: If $R_E$ be the radius of Earth, then the ratio between the acceleration due to gravity at a depth $r$ below and a height $r$ above the Earth surface is (Given, $r<R_E$)

Imagine you are standing on the surface of the Earth, which has a radius of $R_E$. The problem asks us to explore how the acceleration due to gravity changes as we move away from the surface in two opposite directions.

First, we dig a tunnel and go down to a depth $r$. Second, we take a rocket and fly up to a height $r$. Our mission is to find the exact ratio of the gravity at this depth, $g_d$, to the gravity at this height, $g_h$.

To solve this, we need to recall our fundamental formulas for the variation of gravity.

For a point inside the Earth at a depth $r$, the gravity decreases linearly because the mass of the outer shell no longer contributes to the net gravitational pull. The exact formula is:

For a point outside the Earth at a height $r$, gravity follows the inverse-square law. The exact formula is:

Now, we set up the ratio $\frac{g_d}{g_h}$. When we divide the two expressions, the surface gravity $g$ cancels out perfectly. The squared term in the denominator of $g_h$ flips up to the numerator, giving us:

Let's expand the squared bracket first using the standard identity $(a+b{)}^2=a^2+2ab+b^2$:

Next, we carefully multiply this expanded quadratic expression by the linear term $\left(1-\frac{r}{R_E}\right)$:

Distributing the terms, we get:

Finally, combining the like terms yields our elegant final answer:

Why Not Approximate?

You might be wondering, "Why didn't we just use the binomial approximation $g_h\approx g\left(1-\frac{2r}{R_E}\right)$?"

That is a brilliant question! The approximation is only valid when $r\ll R_E$. However, if you look closely at the options provided in the question, they explicitly contain higher-order terms like $r^2$ and $r^3$. This is a massive hint from the examiner that they want the exact algebraic expansion, not an approximation. Always let the options guide your mathematical approach!