Dimensionless Ratios and Numbers
A quantity is dimensionless when its dimensional formula is M^0L^0T^0I^0Theta^0N^0J^0; plane angle and solid angle are dimensionless derived quantities with the special units radian and steradian.
Why this shows up in the exam
Similarity analysis · Material strain reporting · Optical refractive-index comparison
Learn the idea
A dimensionless quantity has zero net exponent for every base dimension, though it may still carry a special unit name. Ratios such as strain or refractive index compare like quantities, so their units cancel and the result can enter universal functions or similarity criteria.
🧠 Memory hook: All base exponents cancel, but a descriptive unit name may remain.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- [x/y] = 1 when [x]=[y] — ratio of like-dimensional quantities
- [strain] = [Delta L/L] = 1 — strain is dimensionless
- [n] = [c/v] = 1 — refractive index is dimensionless
How to approach it
- 1Expand numerator and denominator
- 2Cancel every base exponent
- 3Distinguish pure ratios from radian or steradian labels
Common slip-ups that cost marks
- •Calling every quantity with no written unit dimensionless
- •Treating radians as an independent dimension
- •Assuming all dimensionless quantities have the same meaning
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A quantity Q is defined as Q = force^1 / speed^2. Which dimensional formula represents Q?
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