SI units

Fundamental (base) units measure the seven basic physical quantities that cannot be broken down into anything simpler:

Derived units are built from these seven by substituting into the defining equation of a quantity. Some are used so often that they earn a special one-word name — force (newton, N), pressure (pascal, Pa), energy (joule, J), power (watt, W), frequency (hertz, Hz), charge (coulomb, C), resistance (ohm, $\Omega$) and potential difference (volt, V) — while most, like speed in $\text{m/s}$ or volume in $\text{m}^3$, simply combine the base units without a special name of their own. The rule that generates every derived unit is the same one that generates every derived quantity in mechanics:

where: the powers $a, b, \dots$ come directly from the algebraic formula defining the quantity — for speed $= \text{distance}/\text{time}$, the unit is $\text{m}^1\text{s}^{-1}$, written $\text{m/s}$; for volume $= l \times b \times h$, it is $\text{m}^3$. Two older systems are named in the syllabus only so you can recognise them: the CGS system (centimetre, gram, second) and the FPS system (foot, pound, second). Both are legitimate systems of units, but neither is SI. Unless a question explicitly tells you otherwise, every measurement and every final numerical answer in ICSE Class 9 Physics must be reported in SI units — mixing cgs, fps and SI inside one calculation is the single commonest source of a wrong numerical answer.

Question: A rectangular block has length $l = 25\text{ cm}$, breadth $b = 0.4\text{ m}$, and height $h = 5\text{ mm}$. Express all three measurements in SI units, find the SI unit its volume should be reported in, compute the volume, and if the block has mass $1.5\text{ kg}$, find its density in SI units.

Step 1. Convert each length to the SI unit of length, the metre: $l = 25\text{ cm} = 25 \times 10^{-2}\text{ m} = 0.25\text{ m}$; $b = 0.4\text{ m}$ (already SI); $h = 5\text{ mm} = 5 \times 10^{-3}\text{ m} = 0.005\text{ m}$.

Step 2. Identify the SI unit of volume from its defining formula, $V = l \times b \times h$. Since each length is now in metres, three metres multiplied together give $\text{m}^3$ — the correct SI (derived) unit for volume, with no special name of its own.

Step 3. Compute the volume: $V = 0.25 \times 0.4 \times 0.005 = 5 \times 10^{-4}\text{ m}^3$.

Step 4. Find density from its defining formula, $\rho = m/V$, substituting the SI unit of mass (kg) and the SI unit of volume ($\text{m}^3$) found in Step 2: $\rho = \dfrac{1.5\text{ kg}}{5 \times 10^{-4}\text{ m}^3} = 3000\text{ kg/m}^3$.

Answer: $l = 0.25\text{ m}$, $b = 0.4\text{ m}$, $h = 0.005\text{ m}$; volume $= 5 \times 10^{-4}\text{ m}^3$; density $= 3000\text{ kg/m}^3$, the SI unit of density.

• Writing unit symbols with the wrong case, such as "Kg" instead of "kg", or writing out "Newton" with a capital letter instead of using the symbol N (only the symbol is capitalised for units named after people, never the spelled-out word).
• Forgetting to convert cm, mm or km to metres before substituting into a formula — mixing unit prefixes inside one calculation is the commonest source of a wrong final answer.
• Assuming every derived quantity needs its own special unit name; most, like speed ($\text{m/s}$) or volume ($\text{m}^3$), simply combine base units directly.
• Confusing "SI system" with "metric system" — cgs is also a metric system, but it is not SI, and the syllabus expects SI unless told otherwise.

Ask the tutor: Want more practice converting between unit prefixes or building a derived unit's SI form from its formula? Ask the AI tutor for extra examples on SI units.