Dimensional Analysis
A student writes the dimensional formula of the universal gravitational constant G while analysing planetary motion in a project report. Which of the following correctly represents its dimensions?
Select the correct option:
Solution
[M−1L3T−2]
As stated in NCERT Class 11, Chapter 2 (Units and Measurements), every physical constant carries dimensions that can be derived from the defining equation in which it appears. Newton's law of gravitation states F = G m_1 m_2 / r^2, so rearranging gives G = F r^2 / (m_1 m_2). Substituting dimensions: force F has [M^{1}L^{1}T^{-2}], r^2 has [L^{2}], and the mass product has [M^{2}]. Therefore G = [M^{1}L^{1}T^{-2}][L^{2}] / [M^{2}] = [M^{-1}L^{3}T^{-2}]. The option [M^{1}L^{2}T^{-2}] is wrong because it matches energy, not G. The option [M^{-1}L^{2}T^{-1}] is wrong as the length and time powers do not follow from the formula. The option [M^{1}L^{3}T^{-2}] is wrong because the mass power must be negative once the mass product in the denominator is divided out. A quick sanity check confirms G acts to convert mass and distance into force, so an inverse mass dependence is physically expected.
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About This Question
- Subject
- physics
- Chapter
- physics and measurement
- Topic
- dimensional analysis
- Difficulty
- Medium
- Year
- 2025
Solution
Correct Answer:
[M−1L3T−2]
As stated in NCERT Class 11, Chapter 2 (Units and Measurements), every physical constant carries dimensions that can be derived from the defining equation in which it appears. Newton's law of gravitation states F = G m_1 m_2 / r^2, so rearranging gives G = F r^2 / (m_1 m_2). Substituting dimensions: force F has [M^{1}L^{1}T^{-2}], r^2 has [L^{2}], and the mass product has [M^{2}]. Therefore G = [M^{1}L^{1}T^{-2}][L^{2}] / [M^{2}] = [M^{-1}L^{3}T^{-2}]. The option [M^{1}L^{2}T^{-2}] is wrong because it matches energy, not G. The option [M^{-1}L^{2}T^{-1}] is wrong as the length and time powers do not follow from the formula. The option [M^{1}L^{3}T^{-2}] is wrong because the mass power must be negative once the mass product in the denominator is divided out. A quick sanity check confirms G acts to convert mass and distance into force, so an inverse mass dependence is physically expected.
This medium difficulty physics question is from the chapter physics and measurement, covering the topic of dimensional analysis. It appeared in the 2025 exam.
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