Physics · Units And Measurements · NEET
No. Dimensionless and unitless are not the same thing. A quantity can have a unit but still have no dimension. The clearest NEET example is plane angle: it is measured in radian (a unit), but radian = arc length / radius = length / length, so its dimension is M^0 L^0 T^0. So angle has a unit but no dimension. Truly unitless quantities like strain or refractive index are also dimensionless, but the reverse is not always true. Remember the exact NEET line: plane angle and solid angle have units but no dimensions.
Strain = change in length / original length = length / length. When you divide length by length, the L cancels, giving M^0 L^0 T^0. Because both the top and bottom are the same kind of quantity, all dimensions cancel. Strain also has no unit, so it is both dimensionless and unitless. The same logic makes any ratio of two same-type quantities dimensionless.
Refractive index n = speed of light in vacuum / speed of light in the medium = speed / speed. Speed divided by speed cancels the dimension L T^-1 completely, leaving M^0 L^0 T^0. NCERT lists refractive index as a pure number that is the ratio of two similar quantities. So it is dimensionless and unitless.
Yes. Pure numbers such as pi, e, 2, and 1/2 that appear in formulas are dimensionless constants. They are just numbers with no physical dimension. But be careful: not every physical constant is dimensionless. Universal constants like G (gravitational constant) or h (Planck's constant) do have dimensions. Only pure numbers and certain ratios are dimensionless.
NCERT states that the arguments of trigonometric, logarithmic and exponential functions must be dimensionless. You cannot take sin(5 metre) or e^(3 second) because these functions are defined only for pure numbers. This is a common NEET trick: if you see sin(kt) or e^(-at), then kt and at must each be dimensionless, which lets you find the dimensions of k or a.
Plane angle and solid angle have:
The quantities which have the same dimensions as those of solid angle are:
Try the real previous-year questions from this chapter — each with the answer and a full solution.
It is M^0 L^0 T^0. Every base dimension is raised to the power zero, so the quantity has no dimension. Examples are strain, refractive index, relative density, angle and coefficient of friction.
Strain, refractive index, coefficient of friction, plane angle (radian), and solid angle (steradian). Others include relative density, dielectric constant, Poisson's ratio and any pure number like pi. Many are ratios of two same-type quantities.
Yes. Coefficient of friction = friction force / normal force = force / force. Force divided by force cancels completely, giving M^0 L^0 T^0. It is both dimensionless and unitless.
No. All unitless quantities are dimensionless, but not all dimensionless quantities are unitless. Plane angle and solid angle are dimensionless yet carry units (radian and steradian). This is a frequent NEET point.
Dimensional analysis cannot find pure numbers like 2, 1/2 or pi because these have no dimension to track. So even if an equation is dimensionally correct, a missing or wrong dimensionless constant can still make it numerically wrong. This is a key limitation.
