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MCQEasyJEE 2025Dimensions & Dimensional Analysis

JEE Physics 2025 Question with Solution

Match the LIST-I with LIST-II

LIST-I LIST-II

A. Gravitational constant I. [LT−2][LT^{-2}]

B. Gravitational potential energy II. [L2T−2][L^2T^{-2}]

C. Gravitational potential III. [ML2T−2][ML^2T^{-2}]

D. Acceleration due to gravity IV. [M−1L3T−2][M^{-1}L^3T^{-2}]

Choose the correct answer from the options given below :

  • A

    A-IV, B-III, C-II, D-I

  • B

    A-III, B-II, C-I, D-IV

  • C

    A-II, B-IV, C-III, D-I

  • D

    A-I, B-III, C-IV, D-II

Answer

Correct answer:A

Step-by-step solution

Standard Method

Given: Match each quantity in LIST-I with its dimensional formula in LIST-II.

Find: The correct matching among the four options.

Use the formulas for gravitational constant, gravitational potential energy, gravitational potential, and acceleration due to gravity to derive their dimensional formulas.

For gravitational constant GG, from Newton's law of gravitation:

F=Gm1m2r2F = \frac{G m_1 m_2}{r^2}

So,

G=Fr2m1m2G = \frac{F r^2}{m_1 m_2}

Hence,

[G]=[MLT−2][L2][M2]=[M−1L3T−2][G] = \frac{[MLT^{-2}][L^2]}{[M^2]} = [M^{-1}L^3T^{-2}]

So, A \rightarrow IV.

For gravitational potential energy:

U=mghU = mgh

Therefore,

[U]=[M][LT−2][L]=[ML2T−2][U] = [M][LT^{-2}][L] = [ML^2T^{-2}]

So, B \rightarrow III.

For gravitational potential:

V=Um=GMrV = \frac{U}{m} = \frac{GM}{r}

Thus,

[V]=[ML2T−2][M]=[L2T−2][V] = \frac{[ML^2T^{-2}]}{[M]} = [L^2T^{-2}]

So, C \rightarrow II.

For acceleration due to gravity:

g=Fm=GMr2g = \frac{F}{m} = \frac{GM}{r^2}

Hence,

[g]=[LT−2][g] = [LT^{-2}]

So, D \rightarrow I.

Therefore, the correct matching is A-IV, B-III, C-II, D-I, so the correct option is A.

Direct Dimensional Matching

Given: Four gravitational quantities and four dimensional formulas.

Find: The correct option by quick dimensional identification.

Recognize each quantity directly:

  • Gravitational constant GG has dimensions [M−1L3T−2][M^{-1}L^3T^{-2}], so A \rightarrow IV.
  • Potential energy is energy, so its dimensions are [ML2T−2][ML^2T^{-2}], so B \rightarrow III.
  • Gravitational potential is energy per unit mass, so its dimensions are [L2T−2][L^2T^{-2}], so C \rightarrow II.
  • Acceleration due to gravity is an acceleration, so its dimensions are [LT−2][LT^{-2}], so D \rightarrow I.

This gives A-IV, B-III, C-II, D-I. Therefore, the correct option is A.

Common mistakes

  • Confusing gravitational potential with gravitational potential energy. Potential is energy per unit mass, so its dimensions are [L2T−2][L^2T^{-2}], not [ML2T−2][ML^2T^{-2}]. First identify whether the quantity is total energy or energy per unit mass.

  • Using the dimensions of force incorrectly while deriving GG. From F=Gm1m2r2F = \frac{Gm_1m_2}{r^2}, the factor r2r^2 must go to the numerator when solving for GG. Otherwise the power of LL becomes wrong.

  • Treating acceleration due to gravity gg as if it were a force. gg is acceleration, so its dimensions are [LT−2][LT^{-2}]. Use F=mgF = mg to divide force by mass, not to assign force dimensions directly to gg.

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