Units and Measurements

Written and reviewed by Akanksha Tiwari, Physics & Chemistry faculty at UniDrill (M.Sc. Physics, Banaras Hindu University · 8 years teaching Class XI–XII science). Based on NCERT Physics Class XI, Chapter 1.

Why is 2 m not the same as 2.000 m?

A tailor tears off "two metres" of cloth by eye. A lab technician reports 2.000 m, certified to a tenth of a millimetre. The number is the same, but the claim is not. Every measurement in physics is a number, a unit and a promise about how far you can trust the number. This chapter teaches all three: the units the world agrees on, the digits you are allowed to write, and the errors you must admit to.

Units: agreeing on the yardstick

To measure is to compare. You compare the quantity with an agreed reference called a unit, and the answer is "so many units". A small set of base quantities is chosen, and every other quantity is built from them. The units of the base quantities are base units. Units of everything else are derived units. Together they make a system of units.

Old systems were CGS (cm, g, s), FPS (foot, pound, second) and MKS (m, kg, s). Today the world uses the SI (Système International d'Unités), which is decimal, so converting between a kilometre and a millimetre is only a matter of powers of ten. SI has seven base quantities.

The ampere, kelvin, mole and candela are easy to forget because they are not the everyday ones. A coulomb is not a base unit: it is derived (1 C = 1 A s). When you use the mole, always say what you are counting: atoms, molecules, ions or electrons.

Beyond the textbook. For the exam: each SI base unit is defined by fixing the numerical value of a natural constant, so the metre comes from c, the kilogram from h, the second from the caesium frequency, and so on. In reality: until 20 May 2019 the kilogram was the mass of a single platinum-iridium cylinder kept near Paris, and comparisons with its official copies showed small drifts over the decades. Fixing h removed the dependence on one physical object. Why it matters: a unit tied to a constant of nature is the same on every planet and cannot be damaged.

Two angle units sit outside the seven. The radian measures a plane angle, $d\theta = ds/r$ (arc divided by radius). The steradian measures a solid angle, $d\Omega = dA/r^2$ (area divided by radius squared). Each is a ratio of two like quantities, so both are dimensionless. They are traditionally called supplementary units, and they are never base units.

Derived units get their own names when they are used often: newton (N), joule (J), watt (W), pascal (Pa), hertz (Hz). Very large and very small values use prefixes.

Some non-SI units are still used because the SI ones are inconvenient at those scales.

Significant figures: the digits you may write

A measurement has some digits you are sure of and one last digit that is doubtful. Together they are the significant figures. If a stopwatch shows 1.62 s, the 1 and 6 are reliable, the 2 is the guess, and you must not write 1.620 s because that would claim a precision you do not have.

The number of significant figures depends on the instrument, never on the unit you choose. So 2.308 cm, 0.02308 m, 23.08 mm and 23 080 μm all have four significant figures.

Pause and predict: a student records one length as 7.0 cm and another as 7.00 cm. Which instrument is more precise?

Check your answer: the one that gave 7.00 cm. It claims the hundredths place, so its least count is smaller. Precise does not mean accurate, though. Both readings could still be off from the true length if the instrument has a zero error.

Arithmetic with measured numbers. The answer can never be more precise than the weakest input, but the two operations measure "weakest" differently.

Worked example: adding masses. Add 436.32 g, 227.2 g and 0.301 g, and report the total correctly.

The common wrong step: reporting 664 g because the "least number of significant figures" rule was used. That rule belongs to multiplication and division. For a sum, 227.2 g is uncertain in the tenths place, so nothing beyond the tenths place in the answer is trustworthy.

A related surprise: subtraction can destroy significant figures. 12.9 g − 7.06 g = 5.84 g on a calculator, but the answer is 5.8 g because 12.9 has only one decimal place.

Worked example: volume of a block. A metal block measures 12.5 cm × 3.2 cm × 0.85 cm. Find its volume.

The common wrong step: writing 34.0 cm³ because the calculator showed it. That claims three significant figures when two of your three measurements have only two.

Rounding off. Look at the first digit you are dropping. If it is more than 5, raise the last kept digit by one. If it is less than 5, leave it. If it is exactly 5 (nothing after it), raise the last kept digit only when it is odd. So 2.745 → 2.74, 2.735 → 2.74, and 2.746 → 2.75. In a long calculation, keep one extra digit in the middle and round only at the end.

Beyond the textbook. For the exam: use the odd/even rule above. In reality: this is called round-half-to-even, and it is the default rounding mode in standard computer arithmetic. Always rounding a 5 upward pushes a long column of numbers slightly high, and the odd/even rule cancels that bias. Why it matters: it is not an arbitrary rule, which makes it easier to remember.

Order of magnitude. Write the quantity as $a \times 10^{b}$ with $1 \le a < 10$. If $a \le 5$ the order of magnitude is $10^{b}$, and if $a > 5$ it is $10^{b+1}$. A hydrogen atom of radius $5 \times 10^{-11}$ m is of order $10^{-11}$ m, and Earth's mass of $5.97 \times 10^{24}$ kg is of order $10^{25}$ kg.

Errors: admitting what you don't know

No measurement is perfect. Accuracy is how close a reading is to the true value. Precision is how closely repeated readings agree with each other. A watch that is always exactly 5 minutes fast is precise but not accurate.

For repeated readings $a_1, a_2, \ldots a_n$, the best estimate is the mean $\bar{a}$. The absolute error of each reading is its deviation from the mean, and the mean absolute error is the average of those deviations. Relative error is $\Delta\bar{a}/\bar{a}$, and percentage error is that times 100 %.

Worked example: reporting a measurement. Four readings of the length of a wire are 20.0, 20.4, 19.6 and 20.0 cm. Report the length with its error.

Combining errors. Add the errors of two measured quantities, but add them in the right form.

Notice that in a difference the errors still add. That is why subtracting two nearly equal quantities is dangerous: the answer shrinks but the error does not, so the percentage error explodes. The 80 ± 2 above has a 2.5 % error, while each input had only 1 % and 5 %.

Worked example: percentage error in density. A cylinder has mass m = 50.0 ± 0.5 g, radius r = 2.00 ± 0.02 cm and length l = 10.0 ± 0.1 cm. Find the percentage error in its density $\rho = m/(\pi r^2 l)$.

The common wrong step: forgetting the factor 2 for $r^2$ (giving 3 %) or subtracting the errors of r and l because they are in the denominator. Errors never cancel, so the sign of the power does not matter, only its size.

Dimensions: what kind of quantity is it?

The dimensions of a quantity are the powers to which the base quantities are raised to build it. A velocity is length over time whether it is in km h⁻¹ or m s⁻¹, so its dimensional formula is $[L\,T^{-1}]$. Magnitude, direction and unit do not matter. That is why initial velocity, final velocity and change in velocity all have the same dimensions.

Pause and predict: torque and work both have dimensions $[M\,L^2\,T^{-2}]$. Does that mean torque can be measured in joules?

Check your answer: no. Dimensions tell you the kind of building blocks, not the physical meaning. Work is a scalar and torque is a vector ($\vec{r} \times \vec{F}$), so we write torque in N m and keep the joule for energy. Equal dimensions are a necessary condition for two quantities to be the same thing, never a sufficient one.

Dimensional analysis rests on the principle of homogeneity: you can add or subtract only quantities of the same dimensions, so both sides of a true equation, and every term in it, must have the same dimensions. That gives three uses.

Here is how the check works in practice. The equation $v^2 = u^2 + 2ax$ has every term in $[L^2\,T^{-2}]$, so it passes. The equation $v = u + at^2$ fails, because $at^2$ is $[L]$, not $[L\,T^{-1}]$. But $x = ut + at^2$ also passes, even though the true equation has a $1/2$ in front of $at^2$. Dimensions cannot see a pure number. Two more limits: the argument of sin, cos, log and exponential functions must be dimensionless, and an equation with many terms may hold only if each term is homogeneous.

Worked example: converting a unit. Express g = 9.8 m s⁻² in km h⁻².

The common wrong step: multiplying by 3600 once instead of squaring it. The time unit appears with power −2, so its conversion factor must appear with power 2 as well.

Worked example: deriving a formula. The time period T of a mass m on a spring of spring constant k depends on m and k. Find the form of T.

What the method cannot do: tell you that $c = 2\pi$. That number comes from solving the equation of motion (you will do it in Oscillations) or from experiment. The same logic gives the pendulum result $T = c\sqrt{l/g}$, with the mass dropping out because its power comes out as zero.

Beyond the textbook. For the exam: dimensional analysis cannot determine dimensionless constants. In reality: it can still be astonishingly useful. In 1941 the physicist G. I. Taylor reasoned that the radius R of a huge explosion's fireball can depend only on the energy E released, the time t and the air density ρ. Dimensions then force $R \propto (E t^2/\rho)^{1/5}$. A published photograph of the first atomic test, with its time stamp, was enough to estimate its energy, and the result was close to the classified figure. Why it matters: dimensions are a physicist's first sanity check on any new problem.

Exam traps

Formula sheet

60-second revision

Where this chapter connects

• Next: Motion in a Straight Line uses these units for every kinematics problem, so the unit conversions here (km h⁻¹ to m s⁻¹) come back immediately.
• Later in Class XI: Oscillations proves the spring and pendulum periods that you derived here up to the constant $2\pi$, and Gravitation gives G, whose dimensions you tabulated above.
• Class XII link: Dual Nature of Radiation and Matter uses Planck's constant h, whose dimensions equal those of angular momentum.