Fundamental and Derived Units

In physics, we measure various physical quantities. To express these measurements, we use units. These units are standardized so that everyone understands the same magnitude when a measurement is stated. The system of units is broadly categorized into fundamental units and derived units. Understanding this distinction is crucial for comprehending physical laws and solving problems.

Fundamental Units

Fundamental units are the basic units from which all other units are derived. They are independent of each other and cannot be expressed in terms of other units. Think of them as the building blocks of any system of measurement. The most commonly used system today is the International System of Units (SI). The SI system defines seven fundamental units.

The Seven Fundamental SI Units

These seven quantities and their corresponding units form the foundation of the SI system:

  • Length: Measured in meters (m). The meter is defined as the length of the path traveled by light in a vacuum during a time interval of 1/299,792,458 of a second.
  • Mass: Measured in kilograms (kg). The kilogram is defined based on the fixed numerical value of the Planck constant, h, which is 6.62607015 × 10-34 J⋅s (joule-seconds).
  • Time: Measured in seconds (s). The second is defined as the time interval during which an atom in its ground state, between two hyperfine levels of the caesium 133 atom, undergoes 9,192,631,770 transitions.
  • Electric Current: Measured in amperes (A). The ampere is defined based on the fixed numerical value of the elementary charge, e, which is 1.602176634 × 10-19 C (coulombs).
  • Thermodynamic Temperature: Measured in kelvins (K). The kelvin is defined based on the fixed numerical value of the Boltzmann constant, k, which is 1.380649 × 10-23 J/K (joules per kelvin).
  • Amount of Substance: Measured in moles (mol). The mole is defined based on the fixed numerical value of the Avogadro constant, NA, which is 6.02214076 × 1023 mol-1.
  • Luminous Intensity: Measured in candelas (cd). The candela is defined based on the fixed numerical value of the luminous efficacy of monochromatic radiation of frequency 540 × 1012 Hz, which is 683 lm/W (lumens per watt).

These definitions, updated in 2019, are based on fundamental physical constants, making them invariant and universally applicable.

Derived Units

Derived units are those units that can be expressed in terms of fundamental units. They are obtained by multiplying or dividing fundamental units. For example, the unit of velocity is meters per second (m/s), which is derived from the fundamental units of length (meter) and time (second).

Examples of Derived Units

Let's look at some common derived units and how they relate to fundamental units:

  • Area: The area of a rectangle is length × width. So, the unit of area is meter × meter = meter2 (m2).
  • Volume: The volume of a cube is length × width × height. So, the unit of volume is meter × meter × meter = meter3 (m3).
  • Velocity: Velocity is defined as displacement divided by time. The unit of velocity is meters / second (m/s).
  • Acceleration: Acceleration is the rate of change of velocity. The unit of acceleration is (m/s) / s = meters per second squared (m/s2).
  • Force: According to Newton's second law, Force = mass × acceleration. The unit of force is kg × m/s2. This derived unit is given a special name: the Newton (N). So, 1 N = 1 kg⋅m/s2.
  • Work or Energy: Work = Force × distance. The unit of work is N × m = (kg⋅m/s2) × m = kg⋅m2/s2. This derived unit is called the Joule (J). So, 1 J = 1 N⋅m = 1 kg⋅m2/s2.
  • Pressure: Pressure = Force / Area. The unit of pressure is N / m2. This derived unit is called the Pascal (Pa). So, 1 Pa = 1 N/m2 = 1 kg/(m⋅s2).
  • Density: Density = Mass / Volume. The unit of density is kg / m3.

Many other derived units exist, each formed by combinations of fundamental units. Some derived units have been given special names and symbols (like Newton, Joule, Pascal) for convenience, but they can always be expressed in terms of the fundamental SI units.

Dimensional Analysis

Understanding fundamental and derived units is key to dimensional analysis. Dimensional analysis is a powerful tool used to:

  • Check the consistency of physical equations.
  • Derive relationships between physical quantities.
  • Convert units.

In dimensional analysis, we express each physical quantity in terms of its fundamental dimensions. The primary dimensions are Length (L), Mass (M), and Time (T). Other fundamental dimensions include Electric Current (A), Thermodynamic Temperature (Θ), Amount of Substance (N), and Luminous Intensity (J), though L, M, and T are the most common in mechanics.

Dimensional Formulas

The dimensional formula of a physical quantity is an expression that shows how the dimensions of that quantity are related to the fundamental dimensions.

Examples of Dimensional Formulas:
  • Velocity:

    Unit: m/s

    Dimensions: [L]/[T] = [L1T-1]

    Dimensional Formula: [M0L1T-1] (We include M0 to show that mass is not a factor).

  • Acceleration:

    Unit: m/s2

    Dimensions: [L]/[T2] = [L1T-2]

    Dimensional Formula: [M0L1T-2]

  • Force:

    Unit: Newton (kg⋅m/s2)

    Dimensions: [M] × [L]/[T2] = [M1L1T-2]

    Dimensional Formula: [M1L1T-2]

  • Work:

    Unit: Joule (kg⋅m2/s2)

    Dimensions: [M] × [L2]/[T2] = [M1L2T-2]

    Dimensional Formula: [M1L2T-2]

Shortcut: To find the dimensional formula of any quantity, first write its definition or formula in terms of basic quantities like mass, length, and time. Then, substitute the dimensions of these basic quantities (M, L, T) into the formula. For example, Pressure = Force/Area. We know Force is [M1L1T-2] and Area is [L2]. So, Pressure = [M1L1T-2] / [L2] = [M1L-1T-2].

Systems of Units

While SI is the most prevalent, other systems of units have been used historically and are still encountered in specific contexts.

  • CGS System (Centimeter-Gram-Second): This system uses centimeter for length, gram for mass, and second for time as its fundamental units. Derived units include dyne for force and erg for energy.
  • FPS System (Foot-Pound-Second): This system uses foot for length, pound for mass, and second for time. It is commonly used in the United States and the United Kingdom. The derived unit for force is the pound-force (lbf).
  • MKS System (Meter-Kilogram-Second): This system uses meter for length, kilogram for mass, and second for time. The SI system is an extension of the MKS system, incorporating additional fundamental units.

The SI system is preferred internationally due to its coherence (derived units are formed without numerical factors other than 1) and its basis on fundamental constants, ensuring consistency across different locations and times.

Practical Units

In addition to the standard units, some practical units are used for specific measurements, often related to fundamental or derived units but with convenient scales.

  • Astronomical Unit (AU): The average distance between the Earth and the Sun, approximately 1.5 × 1011 meters. Used for distances within the solar system.
  • Light-Year (ly): The distance light travels in one year in a vacuum. Used for vast interstellar distances. 1 ly ≈ 9.46 × 1015 m.
  • Angstrom (Å): 1 Å = 10-10 meters. Used to measure atomic dimensions.
  • Nanometer (nm): 1 nm = 10-9 meters. Also used for atomic and molecular scales.
  • Micron (µm): 1 µm = 10-6 meters. Used for microscopic objects.
  • Liter (L): A unit of volume, equal to 1 cubic decimeter (dm3) or 0.001 cubic meters (m3). Commonly used for liquids.
  • Torr: A unit of pressure, approximately equal to 1 mmHg (millimeter of mercury). 1 Torr ≈ 133.322 Pa.
  • Electronvolt (eV): A unit of energy, particularly in atomic and nuclear physics. 1 eV ≈ 1.602 × 10-19 Joules.

These practical units are often defined in terms of SI units for consistency.

Prefixes in the SI System

To express very large or very small quantities conveniently, the SI system uses prefixes that are multiplied by the base unit. These prefixes represent powers of 10.

Prefix Symbol Factor
yotta Y 1024
zetta Z 1021
exa E 1018
peta P 1015
tera T 1012
giga G 109
mega M 106
kilo k 103
hecto h 102
deca da 101
(none) (none) 100 = 1
deci d 10-1
centi c 10-2
milli m 10-3
micro µ 10-6
nano n 10-9
pico p 10-12
femto f 10-15
atto a 10-18
zepto z 10-21
yocto y 10-24
Memory Trick for Prefixes: A common mnemonic for the larger prefixes (G, M, k) is "Great Men Know". For smaller prefixes (m, µ, n, p), try "My Micro Nano Pen".

For example, a kilometer (km) is 1000 meters, a megahertz (MHz) is 106 hertz, and a nanosecond (ns) is 10-9 seconds.

Importance of Consistent Units

Using a consistent system of units, like SI, is paramount in science and engineering. Inconsistency can lead to significant errors, as famously demonstrated by the Mars Climate Orbiter mission failure in 1999, where a mix-up between SI units (Newtons) and imperial units (pound-force) in software caused the spacecraft to enter the Martian atmosphere at the wrong angle and disintegrate.

The distinction between fundamental and derived units, along with the use of prefixes and dimensional analysis, provides a robust framework for understanding and quantifying the physical world around us. Mastering these concepts is the first step in tackling any physics problem.

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