JEE Physics 2023 Question with Solution

Given: Two statements about variation of acceleration due to gravity with height and depth.

Find: Which statement is correct.

For height above earth's surface, the solution states:

$$g' = g\left(1 - \frac{2h}{R}\right)$$

So, as $h$ increases, $g'$ decreases.

For depth below earth's surface, the solution states:

$$g' = g\left(1 - \frac{d}{R}\right)$$

So, as $d$ increases, $g'$ also decreases.

Therefore, Statement I is correct.

Now compare the values at equal height and depth, that is, when $h = d$.

At height:

$$g' = g\left(1 - \frac{2h}{R}\right)$$

At depth:

$$g' = g\left(1 - \frac{d}{R}\right)$$

When $h = d$, these two expressions are not equal because the decrease with height has coefficient $2$, while the decrease with depth has coefficient $1$.

Hence, Statement II is incorrect.

The solution explicitly concludes: Statement I is correct and Statement II is incorrect. This corresponds to the option text given as the third option. Although the solution labels it as A, that label is inconsistent with the listed options. Based on the option texts, the defensible correct choice is C.

Given: Variation of gravity with height and depth.

Find: Whether equal values of $h$ and $d$ imply equal gravitational acceleration.

Inside the earth, the variation is linear:

$$g' = g\left(1 - \frac{d}{R}\right)$$

Above the surface, the variation is faster and is approximated by:

$$g' = g\left(1 - \frac{2h}{R}\right)$$

Now set

$$h = d$$

Then the two expressions become

$$g'_{\text{height}} = g\left(1 - \frac{2h}{R}\right)$$

and

$$g'_{\text{depth}} = g\left(1 - \frac{h}{R}\right)$$

These are unequal for any nonzero value of $h$.

So the same distance upward and downward from the surface does not produce the same value of gravitational acceleration.

Therefore, the correct option is C.