Units and Measurements Notes

A measurement pairs a numerical value with a unit: 0.75 m and 75 cm describe the same length. Changing the unit changes the number, not the physical quantity.

The seven SI base quantities and units are length (metre, m), mass (kilogram, kg), time (second, s), electric current (ampere, A), thermodynamic temperature (kelvin, K), amount of substance (mole, mol), and luminous intensity (candela, cd).

Derived units combine base units. Force has unit kg m s⁻², named the newton (N). A radian and a steradian are dimensionless derived units; the separate “supplementary units” class was abolished in 1995. Dimensionless does not mean that the quantity lacks physical meaning.

Notation check: kg is a unit; M is a mass dimension. We use L for length, T for time, I for electric current, Θ for temperature and N for amount of substance in dimensional formulas. Context distinguishes the amount dimension N from the unit symbol N for newton.

Start from a defining equation, replace each quantity with its dimensions, then combine the powers. Speed is distance divided by time, so its dimensions are $[LT^{-1}]$.

Charge has dimensions $[IT]$, so electric field (force/charge) has $[MLT^{-3}I^{-1}]$. From $PV=nR\theta$, the gas constant has $[ML^2T^{-2}N^{-1}\Theta^{-1}]$.

Common mistake: identical dimensions do not make two quantities interchangeable. Torque and energy share dimensions but describe different things.

Terms added together must have matching dimensions. In $s=ut+\tfrac12 at^2$, both terms on the right have dimension L. Dimensional agreement cannot establish the factor ½ or prove the equation physically correct.

State the assumed variables first. If a pendulum period depends only on length $\ell$, mass $m$ and gravitational acceleration $g$, write $P=C\ell^a m^b g^c$. Comparing powers gives $b=0$, $a+c=0$ and $-2c=1$. Thus $P=C\sqrt{\ell/g}$. Dimensions alone cannot find C or its dependence on a dimensionless amplitude.

Arguments of functions such as $\sin(\omega t)$ and $\exp(-t/\tau)$ must be dimensionless. This helps check an equation but does not determine which function is appropriate. When variables share dimensions, dimensionless ratios may remain undetermined.

Leading zeros locate the decimal point; they do not count as significant digits. Zeros between significant digits count, and written trailing decimal zeros convey precision. Thus 0.00620 has three significant figures, while 6.020 has four. Scientific notation removes ambiguity in whole numbers such as 6200.

• Addition and subtraction: round to the least precise decimal place.
• Multiplication and division: use the fewest significant figures among measured inputs.
• Exact counts and exact conversion factors do not limit significant figures.
• Keep extra digits during intermediate steps; round the final result.

Follow the rounding convention specified in the question. Under nearest-even rounding, an exact halfway case ends on an even retained digit: 2.450 to two significant figures becomes 2.4; 2.550 becomes 2.6. But 2.451 becomes 2.5 because it is above halfway.

A systematic effect, such as an incorrect zero setting, can bias readings and is not removed merely by averaging. Random variation causes readings to scatter; repeated independent measurements can reduce uncertainty in their mean. Correct recording mistakes before analysing data.

When an exercise asks for mean absolute deviation, calculate the mean and then average the magnitudes of deviations from that mean:

For a nonzero mean, the corresponding fractional measure is $d/|\bar{x}|$ and the percentage is $100d/|\bar{x}|$. The deviation can be zero; it is never negative. It measures scatter, not a known error from the true value or a universal uncertainty estimate.

Accuracy versus precision: closely grouped readings can still be biased. A small least count describes resolution; calibration, technique and other effects also matter.

These rules concern worst-case error bounds. For $z=a+b$ or $a-b$, the bound is $\Delta z\leq\Delta a+\Delta b$. Subtracting measurements does not mean subtracting uncertainty bounds.

For positive, nonzero measured inputs and small relative errors in $z=C a^p b^q$, a first-order maximum relative-error estimate is:

Use absolute values of the powers, including negative powers from a denominator. An exact C contributes no measurement uncertainty. For density $\rho=m/\ell^3$, add the mass percentage bound and three times the length percentage bound.

This is not the rule for independent standard uncertainties, which use a root-sum-of-squares calculation under the relevant assumptions. Use the kind of uncertainty requested by the problem.

For a direct vernier with $n$ divisions spanning $(n-1)$ main-scale divisions, least count is $1\,\text{MSD}/n$. If 20 vernier divisions span 19 mm and 1 MSD is 1 mm, least count is 0.05 mm.

The observed reading is the main-scale reading immediately before the vernier zero plus the coinciding vernier division number multiplied by the least count.

For a screw gauge, least count = pitch / number of circular divisions. A 0.5 mm pitch and 50 divisions give 0.01 mm. Add the sleeve reading and the circular-scale contribution.

Corrected reading = observed reading − signed zero error. Subtract a positive zero error; add the magnitude of a negative one. Do not assume zero error is absent unless stated or the question asks only for an uncorrected reading.

Write the conversion factor for each base unit and raise it to the power in the formula. Since 1 kg = 1000 g and 1 m = 100 cm, 1 N = 10⁵ dyne and 1 J = 10⁷ erg. Square or cube the length conversion for area or volume.

Useful exact prefixes include milli (10⁻³), micro (10⁻⁶), nano (10⁻⁹) and pico (10⁻¹²). An ångström is 10⁻¹⁰ m. Keep units attached at every step to catch inverted factors.

With tonne, kilometre and hour as base units, 1 N = (10⁻³ tonne)(10⁻³ km)/(hour/3600)² = 12.96 tonne km hour⁻². The numerical value changes because the force unit changes.