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Derived Unit — Definition, Formula & Examples

A derived unit is a unit of measurement made by combining base units through multiplication or division. For example, the metre per second (m/s) for speed comes from dividing a length unit by a time unit.

In a system of units, a derived unit is any unit expressible as a product of integer powers of the base units. In the SI, the seven base units are the metre, kilogram, second, ampere, kelvin, mole, and candela. A derived unit is called coherent when its expression in base units has a numerical factor of exactly 1, as in 1 N=1 kg⋅m⋅s−21\,\text{N} = 1\,\text{kg}\cdot\text{m}\cdot\text{s}^{-2}. The SI gives 22 coherent derived units special names, such as the newton, pascal, joule, watt, and ohm.

Key Formula

[Q]=ma kgb sc Ad Ke molf cdg[Q] = \text{m}^{a}\,\text{kg}^{b}\,\text{s}^{c}\,\text{A}^{d}\,\text{K}^{e}\,\text{mol}^{f}\,\text{cd}^{g}
Where:
  • [Q][Q] = the unit of the quantity Q
  • m,kg,s,A,K,mol,cd\text{m}, \text{kg}, \text{s}, \text{A}, \text{K}, \text{mol}, \text{cd} = the seven SI base units
  • a,b,c,d,e,f,ga, b, c, d, e, f, g = integer exponents (positive, negative, or zero)

Worked Example

Problem: A force of 200 N presses evenly on a flat area of 4 m². Find the pressure, and write its unit using only SI base units.
Step 1: Pressure is force divided by area. Divide the numbers and divide the units.
p=FA=200 N4 m2=50 N/m2p = \frac{F}{A} = \frac{200\,\text{N}}{4\,\text{m}^2} = 50\,\text{N/m}^2
Step 2: Replace the newton with its base-unit form, 1 N=1 kg⋅m/s21\,\text{N} = 1\,\text{kg}\cdot\text{m/s}^2, then cancel one metre.
kg⋅m/s2m2=kgm⋅s2\frac{\text{kg}\cdot\text{m/s}^2}{\text{m}^2} = \frac{\text{kg}}{\text{m}\cdot\text{s}^2}
Step 3: The SI name for N/m2\text{N/m}^2 is the pascal (Pa).
50 N/m2=50 Pa=50 kg⋅m−1⋅s−250\,\text{N/m}^2 = 50\,\text{Pa} = 50\,\text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}
Answer: The pressure is 50 Pa. In base units, that is 50 kg⋅m−1⋅s−250\,\text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}.

Why It Matters

In physics and chemistry, you check a formula by reducing every derived unit to base units. Both sides must match, a method called dimensional analysis. In 1999, NASA lost the Mars Climate Orbiter partly because one piece of software reported impulse in pound-force seconds while another expected newton-seconds.

Common Mistakes

Mistake: Mixing non-base units and calling the result a named SI unit. For example, dividing 200 N by 4 cm² and reporting 50 Pa.
Correction: A pascal is a newton per square metre, so convert first: 4 cm² = 0.0004 m². The pressure is then 200 ÷ 0.0004 = 500,000 Pa, not 50 Pa.

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