Physical Quantity
Definition1
물리량이란, 측정하여 정량화할 수 있는 물질이나 계의 성질을 의미한다.
Explanation
An arbitrary physical quantity is expressed as the product of a numerical value and a 🔒(26/04/19)측정 단위, which may be abbreviated respectively as value and unit. Concretely, for an arbitrary quantity $Z$, its numerical value is denoted by braces $\left\{ Z \right\}$ and its unit by brackets $\left[ Z \right]$.
$$ Z = \left\{ Z \right\} \times [Z] = \left\{ Z \right\} [Z] $$
The product of two quantities $Z$ and $W$ is represented as follows.
$$ ZW = (\left\{ Z \right\} \left\{ W \right\}) \times ([Z][W]) $$
Dimensions
차원 are used to distinguish different physical quantities; quantities having different dimensions are different kinds of quantities. Dimensions express the fundamental character of a quantity and are used to define base quantities, derived quantities, and dimensionless quantities described below. The fundamental dimensions are the following seven, and every physical quantity can be represented as a combination of them.
$$ \text{질량: } \mathsf{M} \quad \text{길이: } \mathsf{L} \quad \text{시간: } \mathsf{T} \quad \text{전류: } \mathsf{I} $$ $$ \text{온도: } \mathsf{\Theta} \quad \text{물질의 양: } \mathsf{N} \quad \text{광도: } \mathsf{J} $$
Units
🔒(26/04/19)단위 are standards for assigning numerical values to physical quantities. They are essential for assessing the magnitude of a quantity.
In summary, dimensions are a concept for distinguishing physical quantities, while units are a concept for assigning and expressing values for quantities.
Base quantities
기본 물리량 are a subset of quantities that cannot be expressed in terms of other quantities. All other quantities can be derived from the base quantities. The international system of quantities, ISQ specifies the following $7$ base quantities.
| Quantity | SI unit | Dimension symbol | ||
|---|---|---|---|---|
| Name | Symbol | Name | Symbol | |
| Length | $l$, $x$, $r$ | metre | $\mathrm{m}$ | $\mathsf{L}$ |
| Time | $t$ | second | $\mathrm{s}$ | $\mathsf{T}$ |
| Mass | $m$ | kilogram | $\mathrm{kg}$ | $\mathsf{M}$ |
| Thermodynamic temperature | $T$ | kelvin | $\mathrm{K}$ | $\mathsf{\Theta}$ |
| Amount of substance | $n$ | mole | $\mathrm{mol}$ | $\mathsf{N}$ |
| Electric current | $i$, $I$ | ampere | $\mathrm{A}$ | $\mathsf{I}$ |
| Luminous intensity | $I_{\mathrm{v}}$ | candela | $\mathrm{cd}$ | $\mathsf{J}$ |
Therefore the dimension of an arbitrary quantity $Z$ can be written as a product of powers of the base quantities as follows.
$$ [Z] = \left[ \mathsf{L} \right]^{\alpha} \left[ \mathsf{T} \right]^{\beta} \left[ \mathsf{M} \right]^{\gamma} \left[ \mathsf{\Theta} \right]^{\delta} \left[ \mathsf{N} \right]^{\epsilon} \left[ \mathsf{I} \right]^{\zeta} \left[ \mathsf{J} \right]^{\eta} $$
The base quantities can be likened, from an algebraic viewpoint, to a basis of a vector space. An arbitrary vector space $V$’s arbitrary vector $v \in V$ can be expressed using a basis $\left\{ e_{i} \right\}$ as follows. $$ v = \sum_{i} c_{i} e_{i} $$ Just as a basis of a vector space can express every vector via combinations, base quantities likewise express all other physical quantities via their combinations.
Logically they can be compared to axioms. In mathematics, an axiom is a proposition accepted as true without being derived or proven from other propositions. While velocity can be described by length and time, and momentum by mass and velocity, base quantities such as mass and time must be understood and accepted as given.
Derived quantities
유도 물리량 are quantities defined as combinations of the base quantities.
Scalars
| Name | Symbol | SI unit | Dimension |
|---|---|---|---|
| 질량 | $m$ | kilogram $\mathrm{kg}$ | $\mathsf{M}$ |
| Time | $t$ | second $\mathrm{s}$ | $\mathsf{T}$ |
| Volume | $V$ | $\mathrm{m}^{3}$ | $\mathsf{L}^{3}$ |
| 밀도 | $\rho$ | $\mathrm{kg/m^{3}}$ | $\mathsf{ML}^{-3}$ |
Vectors
| Name | Symbol | SI unit | Dimension |
|---|---|---|---|
| 위치 | $x$, $\mathbf{x}$ | - | - |
| 속도 | $v$, $\mathbf{v}$ | - | - |
| 가속도 | $a$, $\mathbf{a}$ | - | - |
| 힘 | $F$, $\mathbf{F}$ | - | - |
| 운동량 | $p$, $\mathbf{p}$ | - | - |
Tensors
| Name | Symbol | SI unit | Dimension |
|---|---|---|---|
| Cauchy stress tensor | $\sigma$ | - | - |
| Maxwell stress tensor | $\mathbf{T}$ | - | - |
Dimensionless quantities
무차원량 are quantities whose dimension is $1$, i.e., for which the exponent of every combined dimension is $0$. If $A$ is dimensionless, then
$$ \left[ A \right] = \left[ \mathsf{L} \right]^{0} \left[ \mathsf{T} \right]^{0} \left[ \mathsf{M} \right]^{0} \left[ \mathsf{\Theta} \right]^{0} \left[ \mathsf{N} \right]^{0} \left[ \mathsf{I} \right]^{0} \left[ \mathsf{J} \right]^{0} = 1 $$