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MCQMediumJEE 2023Variation with Altitude & Depth

JEE Physics 2023 Question with Solution

At a certain depth dd below the surface of the Earth, the value of acceleration due to gravity becomes four times that of its value at a height 3R3R above the Earth's surface. Where RR is the radius of Earth (Take R=6400 kmR = 6400 \, \text{km}). The depth dd is equal to:

  • A

    5260 km5260 \, \text{km}

  • B

    640 km640 \, \text{km}

  • C

    2560 km2560 \, \text{km}

  • D

    4800 km4800 \, \text{km}

Answer

Correct answer:A

Step-by-step solution

Standard Method

Given: At depth dd below Earth's surface, acceleration due to gravity is four times its value at height 3R3R above the surface. Also, R=6400 kmR = 6400 \, \text{km}.

Find: The depth dd.

At depth dd, the solution uses:

gd=GMR2(1−dR)g_d = \frac{GM}{R^2}\left(1 - \frac{d}{R}\right)

At height 3R3R above the surface, the distance from Earth's centre is 4R4R, so:

gh=GM(4R)2g_h = \frac{GM}{(4R)^2}

Since the gravity at depth becomes four times the value at that height:

GMR2(1−dR)=4×GM(4R)2\frac{GM}{R^2}\left(1 - \frac{d}{R}\right) = 4 \times \frac{GM}{(4R)^2}

Working and discrepancy note

Cancelling the common factor GMR2\frac{GM}{R^2} gives:

1−dR=141 - \frac{d}{R} = \frac{1}{4}

Therefore,

dR=34\frac{d}{R} = \frac{3}{4}

So,

d=34Rd = \frac{3}{4}R

Substituting R=6400 kmR = 6400 \, \text{km}:

d=34×6400=4800 kmd = \frac{3}{4} \times 6400 = 4800 \, \text{km}

Therefore, the depth is 4800 km4800 \, \text{km}.

The solution states both "The Correct Option is A" and "So, the correct option is (A): 4800 km", but the listed options show 4800 km4800 \, \text{km} as option D, not A. The numerical working clearly gives 4800 km4800 \, \text{km}, so the defensible correct option from the provided options is D.

A second approach in the source writes

1−dR=1161 - \frac{d}{R} = \frac{1}{16}

and then reaches

d=1516R=4800 kmd = \frac{15}{16}R = 4800 \, \text{km}

This algebra is internally inconsistent because 1516×6400=6000\frac{15}{16} \times 6400 = 6000, not 48004800. Hence the correct result is obtained from the first working, and the correct option from the given list is D.

Common mistakes

  • Using the height formula with distance 3R3R from Earth's centre instead of height 3R3R above the surface. This is wrong because the actual distance from the centre becomes R+3R=4RR + 3R = 4R. Always convert height above the surface into distance from the centre before applying the inverse-square law.

  • Equating the gravity at depth directly to the gravity at height instead of four times that value. This misses the statement "becomes four times" and changes the equation completely. First translate the wording carefully into a mathematical relation.

  • Confusing option labels with numerical values when the solution's has a mismatch. This is wrong because the label shown in the solution is inconsistent with the listed options. Trust the physical working and then map the computed value to the actual option list.

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