SI and Derived Units

Every physical quantity is expressed as a numerical value multiplied by a unit. The SI system fixes seven base units from which all mechanical, thermal and electromagnetic units follow. Mastery here means converting cleanly between CGS, SI and practical units, and recognising compound units as products of base powers.

Concept

The seven SI base units are: metre (m, length), kilogram (kg, mass), second (s, time), ampere (A, current), kelvin (K, temperature), mole (mol, amount) and candela (cd, luminous intensity). All other mechanical units are derived:

• Velocity: $\text{m s}^{-1}$
• Acceleration: $\text{m s}^{-2}$
• Force (newton): $1\,\text{N} = 1\,\text{kg m s}^{-2}$
• Energy (joule): $1\,\text{J} = 1\,\text{N m} = 1\,\text{kg m}^2\,\text{s}^{-2}$
• Power (watt): $1\,\text{W} = 1\,\text{J s}^{-1}$
• Pressure (pascal): $1\,\text{Pa} = 1\,\text{N m}^{-2}$

CGS equivalents you must memorise: $1\,\text{dyne} = 10^{-5}\,\text{N}$, $1\,\text{erg} = 10^{-7}\,\text{J}$, $1\,\text{poise} = 0.1\,\text{Pa s}$.

Derivation

Convert $1\,\text{N}$ to dyne. Force $F = m a$, so the dimensional form is $[M L T^{-2}]$. In SI: $1\,\text{kg} \cdot 1\,\text{m s}^{-2}$. In CGS: $g \cdot \text{cm s}^{-2}$.

$1\,\text{N} = 1\,\text{kg}\cdot 1\,\text{m s}^{-2} = (10^{3}\,\text{g})(10^{2}\,\text{cm s}^{-2}) = 10^{5}\,\text{g cm s}^{-2} = 10^{5}\,\text{dyne}.$

The same multiplicative recipe converts any mechanical unit: replace each base unit by its target equivalent raised to the correct power.

JEE Worked Example

Q. A new system of units uses $\alpha = 2\,\text{kg}$ as mass, $\beta = 5\,\text{m}$ as length and $\gamma = 4\,\text{s}$ as time. Express $1\,\text{J}$ in the new units.

Solution. Energy has dimensions $[M L^2 T^{-2}]$. So $1\,\text{J} = 1\,\text{kg m}^2\,\text{s}^{-2} = \frac{1}{\alpha}\cdot\frac{1}{\beta^2}\cdot \gamma^{2} \;\text{(new units)} = \frac{1}{2}\cdot\frac{1}{25}\cdot 16 = 0.32\,\text{new energy units}.$

Traps

• Forgetting that mass exponent in pressure is $+1$ (Pa $= \text{kg m}^{-1}\text{s}^{-2}$).
• Confusing weight (N) with mass (kg) in MCQ options.
• Dropping powers of $10$ during CGS$\leftrightarrow$SI conversions.
• Treating $\text{N m}$ as energy when it represents torque (vector vs scalar context matters).

Key Takeaways

• Seven base units span all of mechanics.
• Always carry units symbolically until the final numerical step.
• Convert unit-by-unit using base equivalents raised to the right power.
• Distinguish identical compound units used for different quantities (e.g. $\text{N m}$).