Animated Solution for Physics - Gravitation: The change in the value of $g$ at a height $h$ above the surface of the earth is the same as at a depth $d$ below the surface of earth. When both $d$ and $h$ are much smaller than the radius of earth, then which one of the following is correct?

This problem is a classic exploration of how Earth's gravity behaves as we move away from its surface—both upwards into the sky and downwards into the crust. It tests our understanding of the approximate formulas for acceleration due to gravity.

Exploring the Extremes Imagine standing on the surface of the Earth, where the acceleration due to gravity is $g$. If you take a hot air balloon up to a height $h$, the gravity you experience drops to a new value, $g_h$. Conversely, if you take an elevator down a deep mine shaft to a depth $d$, the gravity also drops to a new value, $g_d$.

The problem presents a fascinating scenario: the change in gravity is identical in both cases. Mathematically, this means the drop in gravity going up equals the drop in gravity going down:

This elegantly simplifies to $g_h=g_d$. The gravity at height $h$ is exactly the same as the gravity at depth $d$.

The problem explicitly states a crucial constraint: both $d$ and $h$ are much smaller than the radius of the Earth ($R$). This is our green light to use the binomial approximations.

For a small height $h\ll R$, the gravity is:

For a small depth $d\ll R$, the gravity is:

Now, we substitute our tools into the condition $g_h=g_d$:

Let's perform the algebraic execution. First, we divide both sides by $g$ (since $g eq 0$):

Next, we subtract $1$ from both sides:

The negative signs cancel out, leaving us with:

Finally, multiplying both sides by $R$ yields our answer:

$$d=2h$$

Gravity decreases twice as fast when you go up compared to when you go down.

If you climb a mountain of height $h$, you lose a certain amount of gravity. To lose that exact same amount of gravity by digging, you would have to dig a hole twice as deep ($2h$). This happens because moving upwards increases your distance from the entire mass of the Earth (an inverse-square effect), whereas moving downwards only removes the gravitational contribution of the outer shell of mass you've bypassed (a linear effect).