Units, Dimensions and Measurement — NEET Physics

This is the toolkit the rest of Physics runs on. It looks small — 1–2 direct questions — but they are among the most certain marks in the section, and its ideas (SI units, dimensional formulae, error analysis) underpin every numerical you will ever solve. Better still, dimensional analysis lets you reject wrong options and reconstruct forgotten formulas across the whole paper. This chapter builds the toolkit thoroughly, with the rules and worked examples that make it automatic.

1. The SI system: base and derived quantities

Seven base quantities define all of physics; everything else is derived from them.

A quantity's dimensional formula expresses it in powers of M, L, T (and A, K):

$[\text{Force}]=ML{T}^{-2},[\text{Energy}]=M{L}^2{T}^{-2},[\text{Power}]=M{L}^2{T}^{-3}$

2. Dimensional formulae worth memorising

Equal dimensions ≠ same quantity. Work and torque share $M{L}^2{T}^{-2}$ but are physically different — dimensions test consistency, never identity.

3. The three uses of dimensional analysis

1. Check an equation (homogeneity): every term must share the same dimensions. In $s=ut+\frac{1}{2}a{t}^2$, each term is $L$ — consistent.
2. Derive a relation up to a constant: match powers of M, L, T (e.g. deriving the pendulum period's $\sqrt{L/g}$ form).
3. Convert units between systems by tracking the base-unit powers.

Its limits: dimensional analysis cannot find dimensionless constants (like $\frac{1}{2}$ or $2\pi$), cannot handle equations that add unlike terms, and cannot distinguish quantities with identical dimensions. It is a powerful check, not a complete derivation.

Worked example 3.1. Find the dimensions of $G$ from $F=\frac{G{m}_1{m}_2}{r^2}$. $G=\frac{F{r}^2}{m_1{m}_2}=\frac{(ML{T}^{-2})(L^2)}{M\cdot M}=M^{-1}{L}^3{T}^{-2}.$

4. Significant figures and rounding

• Non-zero digits are always significant.
• Leading zeros are not (0.00340 → 3, 4, 0 → 3 sig figs).
• Trailing zeros after a decimal are significant; a trailing zero in a whole number is ambiguous.
• In multiplication/division, the result keeps the fewest significant figures of the inputs; in addition/subtraction, the fewest decimal places.

Worked example 4.1. $2.0\times 3.00=6.00$, but $2.0$ has only 2 sig figs, so the answer is rounded to $6.0$.

5. Accuracy vs precision

• Accuracy — how close a measurement is to the true value.
• Precision — how close repeated measurements are to each other (fine resolution).

A clock 5 minutes fast is precise (consistent) but inaccurate; a scattered set of readings averaging the right value is accurate on average but imprecise. They are independent — an instrument can have either without the other.

6. Error analysis and propagation

• Absolute error $=|\text{measured}-\text{true}|$; relative error $=\frac{\text{absolute}}{\text{true}}$; percentage error $=\text{relative}\times 100$.
• Systematic errors (zero error, calibration) shift every reading the same way — reducible; random errors scatter both ways — reduced by averaging.

Propagation rules:

• Sum/difference: absolute errors add.
• Product/quotient: relative errors add, each weighted by its power:

$\text{If }Z=\frac{A^a{B}^b}{C^c},\frac{\Delta Z}{Z}=a\frac{\Delta A}{A}+b\frac{\Delta B}{B}+c\frac{\Delta C}{C}$

Worked example 6.1. For $Z=A^{2}B/C$ with percentage errors 1%, 2%, 3%: $\frac{\Delta Z}{Z}=2(1\%)+2\%+3\%=7\%.$ The square on $A$ doubles its contribution.

7. Common traps NEET sets here

• Equal dimensions ⇒ same quantity — false (work vs torque).
• Counting leading zeros as significant — they never are.
• Forgetting the power when combining errors — a squared term contributes double.
• Expecting exact constants from dimensional analysis — it can't give the $\frac{1}{2}$ or $2\pi$.
• Confusing accuracy with precision — a precise instrument can still be inaccurate.

8. Memory aids

• "Force = MLT⁻², build the rest" — energy adds an L, power adds a per-T.
• "Leading zeros lead nowhere" — they are never significant.
• "Powers weight the errors" — $\Delta Z/Z$ sums power × fractional error.
• "Accurate = right, Precise = repeatable" — two different virtues.
• "Dimensions check, never confirm constants" — the tool's limit.

9. Exam protocol

1. Memorise the dimensional formulae of force, energy, power, pressure, momentum, $h$, $G$.
2. Use homogeneity to reject dimensionally wrong options fast.
3. Remember dimensions test consistency, not identity.
4. Count significant figures by the zero rules; round to the fewest sig figs (×÷) or decimals (+−).
5. Keep accuracy (closeness to truth) and precision (repeatability) distinct.
6. For products, add fractional errors weighted by their powers.
7. Convert unit prefixes cleanly ($n=10^{-9}$, $\mu =10^{-6}$, $k=10^3$).