NCERT · Ch 1–2

Units, Measurements & Errors

The grammar of every physics number · Class 11 Physics

1Why measurement is the foundation

Physics is the science of quantities, and a quantity is meaningless until you attach a unit and know how much you can trust it. "The rod is 15" says nothing; "15 cm" says something; "15.0 ± 0.1 cm" says everything — value, unit, and the honesty about how well you know it. This chapter is about all three: the units we agree on, the shortcut called dimensional analysis, and the arithmetic of doubt we call error analysis.

None of it is glamorous, but it is where marks are quietly won and lost. A dimensionally wrong formula is certainly wrong; a result quoted to ten digits from a ruler is certainly overconfident. Learn to catch both on sight.

2SI base units & dimensional formulae

The Système International (SI) builds every physical quantity out of just seven base units. Everything else — force, energy, pressure, charge — is a derived combination of these.

The dimensional formula of a quantity records which base units, and to what powers, it is made of — written in square brackets. Most of school physics lives in just the first three, M, L, T. You do not memorise a table; you read the dimensions off the defining equation. Speed is distance over time, so [L T−1]. Acceleration is speed over time again, so [L T−2]. Force is mass times acceleration, so [M L T−2]. Keep pulling that thread:

Reading dimensions off definitions [force] = [M L T−2]  ·  [energy] = [M L2 T−2]  ·  [power] = [M L2 T−3]  ·  [pressure] = [M L−1 T−2]

Energy is force × distance, so multiply [M L T−2] by [L]. Power is energy per time, so divide by [T]. Pressure is force per area, so divide by [L2]. Once you trust this, you never have to rote-learn a dimensional formula again — you derive it in two seconds from what the quantity means.

Remember this

A pure number has no dimensions. Angles, sin θ, ex, refractive index, relative density, strain, the "2" in a formula — all dimensionless. That single fact powers most of the checks below: whatever sits inside a sine, an exponential or a logarithm must be dimensionless.

3Dimensional analysis — three uses, and its limits

The governing rule is the principle of homogeneity: every term added or equated in a physical equation must carry the same dimensions. You cannot add a length to a time any more than you can add apples to hours. From this one rule come three practical powers.

Use 1 — Check an equation

Take v2 = u2 + 2 a s. Left side: [L T−1]2 = [L2 T−2]. Right side, term by term: u2 is [L2 T−2], and a s is [L T−2][L] = [L2 T−2]. Every term matches, so the equation is dimensionally sound. If even one term had disagreed, the formula would be wrong — no exceptions.

Use 2 — Derive a relation (up to a constant)

Suppose the period T of a simple pendulum depends on its mass m, length l and gravity g. Write it as a product of unknown powers and demand that the dimensions balance:

Assume T ∝ ma lb gc [T] = [M]a [L]b [L T−2]c = [Ma Lb+c T−2c]

Now match the powers on both sides. There is no M on the left, so a = 0. There is no L on the left, so b + c = 0. The power of T is 1, so −2c = 1, giving c = −½ and hence b = ½. Assembling:

T ∝ l½ g−½ = √l√g ,  so  T = k √(l ∕ g)

Notice two things at once. The method correctly told us the period does not depend on mass and grows as the square root of length — real physics, extracted from dimensions alone. But it left behind an unknown pure number k. (Experiment or theory later fixes k = 2π.) That missing constant is exactly the method's blind spot.

Use 3 — Convert units

Because a measured quantity is number × unit, and the physical quantity itself does not change when you switch systems, n1 u1 = n2 u2. To convert, express the unit in its base dimensions and let the powers do the bookkeeping — this is how you turn joules into ergs or km h−1 into m s−1 without guessing.

Remember this — the limits

Dimensional analysis cannot: (1) find dimensionless constants like 2π or ½; (2) derive a relation with more than three unknown quantities (too many powers, too few equations); (3) handle sums such as s = ut + ½at2 — it checks each term but cannot build the "+"; (4) touch anything hidden inside sin, log or ex. A dimensionally correct equation is necessary but not sufficient — it can still be physically wrong.

4Significant figures & rounding

Significant figures are the digits in a measurement you actually trust — every certain digit plus the one estimated digit at the end. They are a compact way of stating precision: 2.50 m claims more than 2.5 m, because it asserts you were sure down to the hundredths place.

The counting rules, in order of how often they trip people up:

When you combine measurements, the answer must not pretend to be more precise than its weakest input:

Rounding rule (NCERT convention). If the digit dropped is greater than 5, round up; if less than 5, leave the last kept digit alone. If it is exactly 5 with nothing meaningful after it, round the preceding digit to make it even — so 2.745 → 2.74 but 2.735 → 2.74. This "round half to even" rule keeps a long column of roundings from drifting steadily upward.

5Errors — least count, absolute, relative, percentage

No measurement is exact. The gap between what you read and the true value is the error (not a mistake — an unavoidable limitation). The starting point is your instrument's least count: the smallest quantity it can resolve. A metre scale reads to 1 mm, a vernier caliper to 0.1 mm, a screw gauge to 0.01 mm. The least count is the built-in floor on your certainty — you can never honestly claim to know a single reading better than this.

accurate, low precision centred on truth, but scattered precise, low accuracy tightly grouped, but off-centre
The bullseye (blue) is the true value. Accuracy is closeness to truth; precision is closeness to one another. A systematic error shifts the whole cluster off-centre (right); random error scatters it (left). They are not the same defect.

Errors come in two families. Systematic errors push every reading the same way — a zero-error in a caliper, a stretched tape, a slow clock — and can be corrected once found. Random errors scatter readings up and down unpredictably; you beat them down by repeating the measurement and averaging.

So take n readings and start with the best estimate, the mean amean. Then:

An absolute error of 1 mm is superb on a 2 m wall (0.05%) and dreadful on a 3 mm wire (33%). That is precisely why the relative error, not the absolute one, tells you how good a measurement really is.

◆ Activity — read a vernier caliper

The vernier scale carries 10 divisions in the space of 9 mm on the main scale, so each vernier division is 0.9 mm — a hair short of a millimetre. That mismatch is the whole trick: exactly one vernier line lines up with a main-scale line, and its number tells you the tenths. Reading = main-scale reading + (coinciding division × 0.1 mm). Slide the jaws and watch the highlighted line and the reading update.

6How errors combine (propagation)

You rarely measure the final quantity directly. You measure a mass and a volume, then divide to get density; you measure length and time, then combine to get g. Each ingredient carries its own error, and those errors flow through to the answer by a few clean rules. The safe, exam-standard approach is to assume the errors pile up in the worst case.

Sums and differences → add the absolute errors

If Z = A + B or Z = A − B, then the worst-case uncertainty is

ΔZ = ΔA + ΔB

Note that even for a difference the errors add — they never cancel, because you must protect against the case where one reading was too high and the other too low. This is why subtracting two nearly-equal numbers is dangerous: the absolute error stays fixed while the result shrinks, so the relative error can blow up.

Products, quotients and powers → add the relative errors

For Z = A B, expand the worst case and keep only first-order terms:

Deriving the product rule Z + ΔZ = (A + ΔA)(B + ΔB) = A B + A ΔB + B ΔA + ΔA ΔB

The last term ΔA ΔB is a tiny error-times-error, so we drop it. Subtract Z = AB and divide through by Z = AB:

ΔZZ = ΔAA + ΔBB

The relative errors add. The identical result holds for a quotient Z = A ∕ B. And a power is just repeated multiplication, so a factor An contributes n times its relative error. The master formula for any product of powers:

General propagation for Z = Ap Bq ∕ Cr ΔZZ = p ΔAA + q ΔBB + r ΔCC

Every power becomes a multiplier on that ingredient's relative error — and notice the power of C in the denominator contributes with a + sign too, for the same worst-case reason. The lesson: the quantity raised to the highest power is where your measurement effort should go, because its uncertainty is amplified the most.

Worked example

A simple-pendulum experiment measures length l = 100.0 ± 0.1 cm and period T = 2.00 ± 0.01 s. Using g = 4π2 l ∕ T2, find g and its percentage error.

Value first. With l = 1.000 m and T = 2.00 s, g = 4π2(1.000) ∕ (2.00)2 = 4π2 ∕ 4 = π2 ≈ 9.87 m s−2.

Now the error. Here l appears to the power 1 and T to the power 2, so the power of T becomes a multiplier:

Δgg = Δll + 2 ΔTT = 0.1100.0 + 2 × 0.012.00 = 0.001 + 0.010 = 0.011

So the percentage error is 1.1%, and the absolute error is Δg = 0.011 × 9.87 ≈ 0.11 m s−2. Report g = 9.87 ± 0.11 m s−2 (about 1%).

The teaching point: even though l and T were measured to similar relative precision, the timing error contributes ten times as much to the answer — purely because T is squared. That is exactly why careful experimenters time many swings at once to shrink ΔT ∕ T.

7Common confusions to clear up

8Check yourself

Class 11 Physics · Units, Measurements & Errors · aligned to NCERT Chapters 1–2 · SmartStudy.School