System of Units and SI Units

In physics, we often measure physical quantities. To do this, we need a standard way to express these quantities. This is where the concept of units comes in. A unit is a standard quantity of a particular type, which is used as a reference for measuring other quantities of the same type. For example, the meter is a unit used to measure length.

Fundamental and Derived Units

Physical quantities can be broadly classified into two categories: fundamental quantities and derived quantities.

• Fundamental Quantities: These are the basic quantities that are independent of other quantities and are chosen by convention. Examples include length, mass, and time.
• Derived Quantities: These quantities can be expressed in terms of fundamental quantities. For example, velocity is derived from length and time (velocity = length/time), and acceleration is derived from velocity and time (acceleration = velocity/time).

Similarly, the units used to measure these quantities are also classified as fundamental units and derived units.

• Fundamental Units: These are the units of fundamental quantities. For example, the meter (m) is the unit of length, the kilogram (kg) is the unit of mass, and the second (s) is the unit of time.
• Derived Units: These are the units of derived quantities. They are obtained by multiplying or dividing fundamental units in some combination. For example, the unit of velocity is meters per second (m/s), and the unit of force is kilogram-meter per second squared (kg⋅m/s²), which is also known as the Newton (N).

Various Systems of Units

Historically, several systems of units have been used. The most common ones are:

1. The CGS System (Centimeter-Gram-Second)

This system was developed in France and uses the following fundamental units:

• Length: Centimeter (cm)
• Mass: Gram (g)
• Time: Second (s)

It is also known as the Gaussian system of units. While it was widely used, especially in older scientific literature, it has largely been replaced by the SI system for general use.

2. The FPS System (Foot-Pound-Second)

This system is primarily used in the United States and the United Kingdom. Its fundamental units are:

• Length: Foot (ft)
• Mass: Pound (lb)
• Time: Second (s)

The unit of force in this system is the poundal, and the unit of energy is the foot-poundal. This system is often encountered in engineering contexts in some countries.

3. The MKS System (Meter-Kilogram-Second)

This system uses the following fundamental units:

• Length: Meter (m)
• Mass: Kilogram (kg)
• Time: Second (s)

The MKS system is a precursor to the SI system and forms its basis. It was found to be more practical for electrical and mechanical measurements.

The International System of Units (SI)

The International System of Units, abbreviated as SI (from the French Système International d'Unités), is the modern form of the metric system and is the most widely used system of measurement in the world. It was established by the General Conference on Weights and Measures (CGPM) in 1960. The SI system is based on seven fundamental units, called SI base units, and two supplementary units.

SI Base Units

There are seven SI base units, each corresponding to a fundamental physical quantity:

Definitions of SI Base Units (as of 2019 redefinition)

The definitions of the SI base units have been refined over time to be based on fundamental physical constants, ensuring stability and universality.

• Meter (m): The meter is defined by taking the fixed numerical value of the speed of light in vacuum, c, to be 299,792,458 meters per second. This definition relates the meter to the second and the speed of light.
• Kilogram (kg): The kilogram is defined by taking the fixed numerical value of the Planck constant, h, to be 6.62607015 × 10^-34 joule-seconds. This definition relates the kilogram to fundamental constants of quantum mechanics.
• Second (s): The second is defined by taking the fixed numerical value of the caesium frequency Δν_Cs, which is the unperturbed ground-state hyperfine transition frequency of the caesium 133 atom, to be 9,192,631,770 hertz. This means one second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom.
• Ampere (A): The ampere is defined by taking the fixed numerical value of the elementary charge, e, to be 1.602176634 × 10^-19 coulombs. This definition links the ampere to the charge of a single electron.
• Kelvin (K): The kelvin is defined by taking the fixed numerical value of the Boltzmann constant, k, to be 1.380649 × 10^-23 joules per kelvin. This definition relates temperature to energy at the microscopic level.
• Mole (mol): The mole is defined by taking the fixed numerical value of the Avogadro constant, N_A, to be 6.02214076 × 10²³ reciprocal mole (mol^-1). This means one mole contains exactly 6.02214076 × 10²³ elementary entities, which must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles.
• Candela (cd): The candela is defined by taking the fixed numerical value of the luminous efficacy of monochromatic radiation of frequency 540 × 10¹² Hz, K_cd, to be 683 lumens per steradian. This definition relates the candela to the lumen and the frequency of light.

SI Supplementary Units

The SI also includes two supplementary units, which are dimensionless and can be treated as either base units or derived units depending on the context.

• Plane Angle: Radian (rad). It is defined as the ratio of arc length to radius. In a circle, an angle of 1 radian subtends an arc whose length is equal to the radius of the circle.
• Solid Angle: Steradian (sr). It is defined as the ratio of the surface area on a sphere to the square of its radius. A solid angle of 1 steradian is subtended at the center of a sphere by a surface area equal to the square of the radius.

SI Derived Units

Derived units are formed by combining the base units according to physical laws. For example:

• Area: Square meter (m²)
• Volume: Cubic meter (m³)
• Velocity: Meter per second (m/s)
• Acceleration: Meter per second squared (m/s²)
• Force: Newton (N) = kg⋅m/s²
• Pressure: Pascal (Pa) = N/m² = kg/(m⋅s²)
• Energy: Joule (J) = N⋅m = kg⋅m²/s²
• Power: Watt (W) = J/s = kg⋅m²/s³
• Electric Charge: Coulomb (C) = A⋅s
• Electric Potential: Volt (V) = W/A = J/C = kg⋅m²/(A⋅s³)

SI Prefixes

SI prefixes are used to form decimal multiples and submultiples of SI units. They are used when the unit itself is too large or too small for a particular application.

For example, 1 kilometer (km) is 1000 meters (m), 1 megahertz (MHz) is 10⁶ hertz (Hz), and 1 nanometer (nm) is 10^-9 meters (m).

Importance of SI Units

The SI system is crucial for several reasons:

• Universality: It is recognized and used globally, facilitating international trade, scientific collaboration, and technological development.
• Consistency: Its coherent system of units means that derived units are formed simply by multiplication and division of base units, without numerical factors.
• Accuracy and Precision: The definitions of SI units are based on fundamental physical constants, ensuring high accuracy and stability.
• Practicality: The SI system is comprehensive, covering all fields of measurement, from everyday use to advanced scientific research.

Unit Conversion

Often, you will need to convert a quantity from one unit to another. This is done by multiplying the quantity by a conversion factor, which is a ratio equal to 1.

Example: Convert 5 kilometers to meters.

We know that 1 kilometer = 1000 meters. The conversion factor can be written as (1000 m / 1 km) or (1 km / 1000 m). We choose the factor that cancels out the original unit.

5 km × (1000 m / 1 km) = 5000 m

Example: Convert 2.5 hours to seconds.

We know: 1 hour = 60 minutes, and 1 minute = 60 seconds.

2.5 hours × (60 minutes / 1 hour) × (60 seconds / 1 minute) = 2.5 × 60 × 60 seconds = 9000 seconds.

Dimensional Analysis

Dimensional analysis is a powerful tool for checking the correctness of equations and deriving relationships between physical quantities. It states that an equation must be dimensionally homogeneous, meaning that each term in an equation must have the same dimensions.

The dimensions of fundamental quantities are represented by symbols:

• Length: [L]
• Mass: [M]
• Time: [T]
• Electric Current: [A]
• Thermodynamic Temperature: [Θ] (Theta)
• Amount of Substance: [N]
• Luminous Intensity: [J]

For example, the dimensions of velocity are [L]/[T] or [L T^-1]. The dimensions of force are [M L T^-2].

Example: Check the dimensional consistency of the equation for distance traveled by a uniformly accelerated object: s = ut + ½at², where s is displacement, u is initial velocity, t is time, a is acceleration.

• Dimensions of s: [L]
• Dimensions of ut: [L T^-1] × [T] = [L]
• Dimensions of ½at²: [L T^-2] × [T]² = [L T^-2] × [T²] = [L]

Since all terms have the dimension of length [L], the equation is dimensionally consistent.