Physical Quantities & Measurements – Concise Exam Notes

Physical and Non-Physical Quantities

A physical quantity is any property of a system that can be measured quantitatively with an instrument, yielding a numerical value and a unit, for example length, time, or temperature. Quantities such as love or beauty lack direct measurement, rely on subjective perception, and are therefore termed non-physical.

Base and Derived Quantities

Scientists selected seven mutually independent base quantities: length, mass, time, temperature, electric current, luminous intensity, and amount of substance. Any other measurable property that can be expressed as an algebraic combination of these is a derived quantity. Thus speed depends on distance and time, while density depends on mass and volume.

SI Units

The International System of Units (SI) assigns one base unit to each base quantity: metre, kilogram, second, kelvin, ampere, candela, and mole. Derived units are written through algebraic relations among base units; for example $ext{m s}^{-1}$ for speed and $ext{kg m}^{-3}$ for density. Unit symbols are never pluralised; their full names start with lowercase (except Celsius), and symbols named after people start with uppercase (e.g. N for newton).

Prefixes and Scientific Notation

SI prefixes represent powers of ten to streamline very large or small values, e.g. $1\,\text{km}=10^{3}\,\text{m}$ and $1\,\text{mm}=10^{-3}\,\text{m}$. Scientific notation expresses any number as $a\times10^{n}$ with 1\le a<10, e.g. Earth–Sun distance $1.38\times10^{8}\,\text{km}$ and hydrogen-atom diameter $5.2\times10^{-11}\,\text{m}$.

Instruments for Length

A metre rule reads to $1\,\text{mm}$ (least count). Parallax error arises if the eye is off-axis. Vernier callipers have a least count of $0.1\,\text{mm}$, obtained from the difference between one main-scale division ($1\,\text{mm}$) and one Vernier-scale division ($0.9\,\text{mm}$). Zero error is checked with closed jaws and algebraically added or subtracted. A micrometre screw gauge reads to $0.01\,\text{mm}$; its least count equals screw pitch (usually $0.5\,\text{mm}$) divided by the number of thimble divisions (usually $50$).

Instruments for Volume and Time

Graduated cylinders read liquid volumes at the meniscus level; the difference before and after submerging a solid gives the solid’s volume. For larger objects a displacement can is used, with overflow collected and measured. Time intervals are taken with mechanical or digital stopwatches; a typical analogue least count is $0.1\,\text{s}$, whereas digital models reach $0.01\,\text{s}$.

Instruments for Mass

A physical balance compares an unknown mass with standard masses using the lever principle. Weight (a force) differs from mass but is measured with spring balances; balances must be levelled and calibrated before use.

Measurement Errors and Uncertainty

Human (personal) errors stem from reading technique or reaction time; systematic errors shift all readings equally due to calibration or zero faults; random errors cause unpredictable scatter from environmental fluctuations. Repeating measurements and averaging mitigates random errors; calibration and zero-error corrections limit systematic ones, while proper technique reduces human errors. Any single reading using an instrument with least count $LC$ carries an uncertainty of roughly $\pm LC/2$.

Significant Figures, Precision and Accuracy

All reliably known digits plus the first estimated digit constitute significant figures. Zeros between non-zero digits and after a decimal are significant; leading zeros are not. Precision reflects the closeness among repeated readings and is dictated by instrument resolution. Accuracy describes closeness to the true value and depends on both systematic error and relative uncertainty; more significant figures generally signal higher accuracy.

Rounding Off

When retaining a limited number of significant figures, round to the nearest value: digits 0–4 stay, 6–9 raise the previous digit by one. For a final 5, raise the preceding digit only if it is odd. Example: $2.512\times10^{3}$ rounded to two significant figures becomes $2.5\times10^{3}$, while $4.55\times10^{2}$ becomes $4.6\times10^{2}$.

Least Count Concept

The least count is the smallest subdivision an instrument can reliably measure. A measurement result equals the main-scale reading plus $(\text{scale reading})\times LC$; hence the least count sets the ultimate precision limit.

Core Takeaways

Measurements rely on standardised units, scientific notation, and properly calibrated instruments. Understanding least count, significant figures, and error types enables reliable, precise, and accurate experimental data.