Dimensional variables are those physical quantities which have dimensions of the form [M^a L^b T^c]… {where,M,L,T are fundamental physical quantities which are Mass,Length and Time respectively. And a,b,c are any real numbers} … but are variables.
Relative density, refractive index and Poisson ratio all the three are ratios, therefore they are dimensionless constants.
Dimensional formula is an expression for the unit of a physical quantity in terms of the fundamental quantities. The fundamental quantities are mass (M), length (L), and time (T). A dimensional formula is expressed in terms of powers of M,L and T.
A dimensionless variable (DV) is a unitless value produced by (maybe repeatedly) multiplying and dividing combinations of physical variables, parameters, and constants.
The measured value of the constant is known with some certainty to four significant digits. In SI units, its value is approximately 6.674×10
−11 m
3⋅kg
−1⋅s
−2.
Gravitational constant.
| Values of G | Units |
|---|
| 6.67430(15)×10−11 | m3⋅kg–1⋅s–2 |
| 4.30091(25)×10−3 | pc⋅M⊙–1⋅(km/s)2 |
Those constant which have dimension but constant value called dimensional constants, example of these constant like gravitational constant, electric field constant etc there are many dimension constants .
| Characteristic | |
|---|
| Other Metric (SI) Equivalents with More Basic Units | 8.9875517874 kilometer3 kilograms / ampere2 second4 |
| Metric (SI) Dimensions | length3 × mass × time-4 × electric-current-2 |
| Description in Terms of Other Units and Constants | 1 / (4 pi permittivity-of-vacuum) |
All constants are dimensionless.
Or, C = [M1 L2 T-2] × [M1 L0 T0]-1 × [M0 L0 T0 K1]-1 = [M0 L2 T-2 K-1]. Therefore, specific heat capacity is dimensionally represented as [M0 L2 T-2 K-1].
Therefore, the electric charge is dimensionally represented as [M0 L0 T1 I1].
Or, L = [M1 L2 T-2] × [M1 L0 T0]-1 = [M0 L2 T-2]. Therefore, latent heat is dimensionally represented as [M0 L2 T-2].
Capacitance
| Common symbols | C |
|---|
| Other units | μF, nF, pF |
| In SI base units | F = A2 s4 kg−1 m−2 |
| Derivations from other quantities | C = charge / voltage |
| Dimension | M−1 L−2 T4 I2 |
Gravitational constant also known as universal gravitational constant has a symbol G and has a dimension [M−1L3T−2] while others are dimensionless constant.
| Click symbol for equation |
|---|
| Wien wavelength displacement law constant† |
|---|
| Numerical value | 2.897 771 955 x 10-3 m K |
| Standard uncertainty | (exact) |
| Relative standard uncertainty | (exact) |
Or, E = [M] × [L1 T-1]2 = M1 L2 T-2. Therefore, energy is dimensionally represented as M1 L2 T-2.
An angle symbolically has dimension . For consistency in the Units package, angles have the dimension length/length(radius). The SI derived unit of angle is the radian, which is defined as the angle for which the radius equals the arclength.
The dimension of Planck's constant is the product of energy multiplied by time, a quantity called action. Planck's constant is often defined, therefore, as the elementary quantum of action. Its value in metre-kilogram-second units is defined as exactly 6.62607015 × 10−34 joule second.
The potential difference is the work done per unit charge. The dimensional formula for potential difference is dimension of work/dimension of charge =dimensions of mass times acceleration times distance/ dimension for charge =M^1L^1T^-2L^1 / A^1T^1 = M^1 L^2 T^-3A^-1 HENCE OPTION "C" IS CORRECT.
Or, T = [M1 L1 T-2] × [L-1] = M1 T-2. Therefore, surface tension is dimensionally represented as M1 T-2.
As part of the 2019 redefinition of SI base units, the Boltzmann constant is one of the seven "defining constants" that have been given exact definitions. They are used in various combinations to define the seven SI base units. The Boltzmann constant is defined to be exactly 1.380649×10−23J⋅K−1.
Units and dimensions
| Quantity | Dimension | Unit |
|---|
| acceleration | [L T-2] | meter per second squared |
| density | [M L-3] | kilogram per cubic meter |
| force | [M L T-2] | newton |
| pressure | [M L-1 T-2] | pascal |
In classical statistical mechanics, Boltzmann Constant is used to expressing the equipartition of the energy of an atom. It is used to express Boltzmann factor. It plays a major role in the statistical definition of entropy. In semiconductor physics, it is used to express thermal voltage.
The dimension of thermal conductivity is M1L1T−3Θ−1, expressed in terms of the dimensions mass (M), length (L), time (T), and temperature (Θ).
Therefore, frequency is dimensionally represented as [M0 L0 T1].
Frequency, in physics, the number of waves that pass a fixed point in unit time; also, the number of cycles or vibrations undergone during one unit of time by a body in periodic motion.
Frequency is the number of completed wave cycles per second.This frequency definition leads us to the simplest frequency formula: f = 1 / T . f denotes frequency and T stands for the time it takes to complete one wave cycle measured in seconds.
Frequency is equal to 1 divided by the period, which is the time required for one cycle. The derived SI unit for frequency is hertz, named after Heinrich Rudolf Hertz (symbol hz). One hz is one cycle per second.
Or, T = √[M0 L1 T0] × [M0 L1 T-2]-1 = √[T2] = [M0 L0 T1]. Therefore, the time period is dimensionally represented as [M0 L0 T1].
Density = Mass × Volume-1. Or, (ρ) = [M1 L0 T0] × [M0 L3 T0]-1 = [M1 L-3 T0] Therefore, density is dimensionally represented as [M1 L-3 T0].
The dimension of force constant can be operated using the spring force formula i.e. F = -Kx. It gives k = -F/x. The SI unit of spring constant is N.m?¹.
[2πr]=[2]⋅[π]⋅[r]=1⋅1⋅L=L, since the constants 2 and π are both dimensionless and the radius r is a length. We see that 2πr has the dimension of length, which means it cannot possibly be an area.
