Dimensional Analysis

Question

An engineer notes that pressure and energy density appear in the same equation while studying a compressed gas cylinder. Which statement about their dimensional relationship is correct?

RankGuru · Accepted Answer

Following NCERT Class 11, Chapter 2 (Units and Measurements), dimensions are found by reducing each quantity to base quantities. Pressure is force per unit area, giving [M^{1}L^{1}T^{-2}] / [L^{2}] = [M^{1}L^{-1}T^{-2}]. Energy density is energy per unit volume, giving [M^{1}L^{2}T^{-2}] / [L^{3}] = [M^{1}L^{-1}T^{-2}]. Since both reduce to the same expression, pressure and energy density share identical dimensions. The claim that pressure has an extra length dimension is wrong because the length powers are equal at minus one. The claim that energy density has an extra mass dimension is wrong since both contain mass to the first power only. The claim that they differ by a time dimension is wrong because both carry the same time power of minus two. A final check confirms this equality is why pressure can be physically interpreted as energy stored per unit volume in a gas, which is conceptually consistent.

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