Question:
The acceleration due to gravity at a height $h$ above the Earth's surface is the same as that at a depth $d$ below the surface. If both $h$ and $d$ are much smaller than the radius of Earth $R$, then the relation between $h$ and $d$ is:
The acceleration due to gravity at a height $h$ above the Earth's surface is the same as that at a depth $d$ below the surface. If both $h$ and $d$ are much smaller than the radius of Earth $R$, then the relation between $h$ and $d$ is:
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Near the surface, gravity decreases twice as fast as you go upwards compared to going downwards. Therefore, to experience the same reduction in gravity, the depth must be twice the height ($d = 2h$).
Updated On: May 31, 2026
- $d = 2h$
- $h = 2d$
- $d = h$
- $d = 4h$
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The Correct Option is A
Solution and Explanation
Step 1: Concept
For small altitudes $h \ll R$, the acceleration due to gravity at a height is approximated by $g_h = g \left(1 - \frac{2h}{R}\right)$. The acceleration due to gravity at a depth is $g_d = g \left(1 - \frac{d}{R}\right)$.
Step 2: Meaning
We equate the gravity variations for height and depth to find the geometric relationship between $h$ and $d$.
Step 3: Analysis
Equating $g_h$ and $g_d$: \[ g \left(1 - \frac{2h}{R}\right) = g \left(1 - \frac{d}{R}\right) \] Simplifying the terms: \[ 1 - \frac{2h}{R} = 1 - \frac{d}{R} \implies \frac{2h}{R} = \frac{d}{R} \implies d = 2h \]
Step 4: Conclusion
The depth $d$ is related to height $h$ by the relation $d = 2h$. Final Answer: (A)
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