Q.Give an example of

• (a) Angle (radian): $\theta = \text{arc}/\text{radius}$, a ratio of two lengths — dimensionless, but still carries the unit "radian."
• (b) Strain ($\Delta L/L$): ratio of two lengths — no dimension, and conventionally no unit either.
• (c) Planck's constant $h = 6.626 \times 10^{-34}\ \text{J·s}$ — has both a unit (J·s) and a dimension $[ML^2T^{-1}]$. …

Dimensions vs. units — the key distinction

Dimension tells you the physical kind of a quantity (length, mass, time, ...). Unit is simply the scale used to express it. A quantity can be dimensionless (no physical "kind" — it's a pure number) and still be given a unit as a labeling convention.

(a) A physical quantity with a unit but no dimensions

Plane angle. An angle in radians is defined as $\theta = \dfrac{\text{arc length}}{\text{radius}}$ — a ratio of two lengths, so the length dimensions cancel completely: $[\theta] = M^0L^0T^0$ (dimensionless). Yet we still assign it the unit radian. (Solid angle, in steradians, is the analogous example.)

(b) A physical quantity with neither unit nor dimensions

Strain ($\Delta L / L$, the ratio of change in length to original length). Both numerator and denominator are lengths, so the ratio is a pure number: no dimension, and — unlike angle — conventionally reported with no unit at all, just a decimal or a percentage.

(c) A constant which has a unit …

Concept: Dimensions and Units of Physical Quantities

Every physical quantity can be described by:

• Dimensions (expressed in terms of mass $M$, length $L$, time $T$, etc.)
• Units (the standard of measurement, e.g., metre, second, kilogram)

A quantity may have one without the other — this is the core idea.

Method: Classification by Dimensional Formula

Steps:

1. Write the dimensional formula of the quantity using $[M^a L^b T^c]$.
2. Check if all exponents are zero → dimensionless.
3. Check if the quantity has a standard SI unit → has a unit.
4. Combine the two checks to classify into the four categories.

Examples

(a) A physical quantity which has a unit but no dimensions

Example: Angle (radian)

• Dimensional formula: $[M^0 L^0 T^0]$ → dimensionless
• SI unit: radian (a unit exists)

Why? Angle is a ratio of arc length to radius — the units cancel, but we still assign "radian" for clarity.

(b) A physical quantity which has neither unit nor dimensions

Example: Pure number (e.g., $\pi$, $e$, refractive index)

• Dimensional formula: $[M^0 L^0 T^0]$ → dimensionless
• No SI unit assigned

Why? A pure ratio like refractive index = $\frac{\text{speed in vacuum}}{\text{speed in medium}}$ has no unit — it's just a number.

(c) A constant which has a unit

Example: Universal gravitational constant $G$

• Value: $6.67 \times 10^{-11}$
• Unit: $\text{N m}^2/\text{kg}^2$ (or $\text{m}^3 \text{kg}^{-1} \text{s}^{-2}$)
• Dimensions: $[M^{-1} L^3 T^{-2}]$

Why? $G$ appears in $F = \frac{G m_1 m_2}{r^2}$ — the equation forces it to carry units to balance both sides.

(d) A constant which has no unit …

Here are the common mistakes students make on this concept — dimensions and units of physical quantities and constants — and how to avoid each.

(a) A physical quantity which has a unit but no dimensions

✗ Common Mistake

Students often write angle (radian) but then incorrectly say it has no unit — or they write strain and claim it has dimensions.

✓ Correct Understanding

• Angle (radian) is the classic example: it is defined as the ratio of arc length to radius. Both are lengths, so dimensions cancel → dimensionless.
• However, the radian is a unit (SI supplementary unit). So it has a unit but no dimensions.

🛠 How to Avoid

• Remember: dimensionless ≠ unitless. A quantity can be a pure number (dimensionless) but still have a named unit (radian, steradian).
• Strain (change in length / original length) is also dimensionless, but it has no named unit — it’s just a number. So strain is not the answer here.

Correct answer: Angle (measured in radians) or solid angle (steradian).

(b) A physical quantity which has neither unit nor dimensions

✗ Common Mistake

Students write pure numbers like 2, π, or e — but these are constants, not physical quantities. The question asks for a physical quantity.

✓ Correct Understanding

• A physical quantity is something measurable (like length, time, mass). A pure number is not a physical quantity.
• Examples that work: Strain, refractive index, relative density, specific gravity. All are ratios of two same-dimension quantities → dimensionless and unitless.

🛠 How to Avoid

• Distinguish: constant (like π) vs physical quantity (like strain). The question says “physical quantity”.
• If it’s a ratio of two identical physical quantities, it’s dimensionless and unitless.

Correct answer: Strain, refractive index, relative density.

(c) A constant which has a unit

✗ Common Mistake

Students write π or e — but these are dimensionless constants with no unit.

✓ Correct Understanding

• Many fundamental constants have units. For example:
  • Gravitational constant $G$: $\text{N m}^2/\text{kg}^2$
  • Planck’s constant $h$: $\text{J s}$
  • Speed of light $c$: $\text{m/s}$

🛠 How to Avoid

• Check: does the constant’s value change if you change the unit system? If yes, it has a unit.
• Constants like π, e are pure numbers — they are unitless.

Correct answer: $G$, $h$, $c$, $R$ (universal gas constant), etc.

(d) A constant which has no unit

✗ Common Mistake …