Which of the following statements is correct with respect to dimensional analysis?
Show answer & explanation
Correct answer: C
Explanation
Rule of Dimensional Analysis: Mathematical functions such as trigonometric (sin θ, cos θ), logarithmic (log x), and exponential (eˣ) functions are pure numbers. Therefore, the arguments or quantities within these functions must always be dimensionless. For example, in sin(ωt), the dimensional formula of ωt must be [M⁰L⁰T⁰].
Analysis of other options (Why are they incorrect?):-
Option 1 is incorrect: An equation that is dimensionally correct is not necessarily physically correct. For example, the equation of motion s = vt + at² appears dimensionally consistent but is physically incorrect, as the correct equation is s = ut + 1/2 at². Dimensional analysis cannot determine numerical constants (like 1/2).
Option 2 is incorrect: Dimensional analysis can never verify the magnitude of an equation or pure numerical constants.
Option 4 is incorrect: Refractive index (n) is the ratio of two similar quantities:
n = Speed of light in vacuum / Speed of light in medium
The identical units (m/s) cancel out in the ratio. Therefore, the refractive index is a dimensionless and unitless quantity.
Potential questions for upcoming exams:-
Question: What does the Principle of Homogeneity state?
Answer: The dimensions of every term on both sides of any physical equation must be identical. You can only add or subtract physical quantities that have the same dimensions.
Question: Which of the following quantities is dimensionless but has a unit?
Answer: Plane Angle (Unit: Radian) and Solid Angle (Unit: Steradian).