MHT CET · Physics · Gravitation

If the change in the value of $g$ at a height $h$ above the surface of the earth is the same as at a depth $x$ below it, then (both $x$ and $h$ being much smaller than the radius of the earth)

A $x = h$
B $x = 2 x h$
C $x = \frac{h}{2}$
D $x = h^{2}$

Verified Solution

Answer & Solution

Correct Answer

(B) $x = 2 x h$

Step-by-step Solution

Detailed explanation

The value of g at the height h from the surface of earth

g^{'} = g [1 - \frac{2 h}{R}]

The value of g at depth x below the surface of earth

g^{'} = g [1 - \frac{x}{R}]

These two are given equal, hence

[1 - \frac{2 h}{R}] = [1 - \frac{x}{R}]

On solving, we get

x = 2 h

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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If the change in the value of g at a height h above the surface of the earth is the same as at a depth x below it, then (both x and h being much smaller than the radius of the earth)

Step-by-step Solution

If the change in the value of $g$ at a height $h$ above the surface of the earth is the same as at a depth $x$ below it, then (both $x$ and $h$ being much smaller than the radius of the earth)