Subsection 1.1.1 Derived Units
Derived units are the unit of such physical quantities which depends on the base physical quantities via a mathematical relation between them. Some of the derived units are expressed below.
-
Area =\(Length \times Length = L^{2}\)i.e. \(m^{2}\) in SI system, \(cm^{2}\) in CGS system, and \(ft^{2}\) in FPS system.
-
Density =\(\frac{mass}{volume}=\frac{M}{L^{3}}\)i.e. \(kg/m^{3}\) in SI system, \(g/cm^{3}\) in CGS system, and \(lb/ft^{3}\) in FPS system, (lb stands for pound).
-
Speed =\(\frac{length}{time}\)i.e. \(m/s\) in SI system, \(cm/s\) in CGS system, and \(ft/s\) in FPS system.
-
Electric resistance = \(\frac{\text{electric potential difference}}{current}\)i.e. \(V/A = kgm^{2}/A^{2}s^{3}=\Omega (Ohm)\) in SI system.
-
Magnetic flux =electric potential \(\times time\)i.e. \(Vs\) or wb (weber) in SI system.
Subsubsection 1.1.1.1 Dimensional Analysis
We can express the derived physical quantity by the dimensions of the base quantities. The dimensions of mass, length, and time are written as [M], [L], and [T], respectively. The square brackets round the letter represent dimensions of a base quantity. The dimensions of any other quantity will involve one or more of these basic dimensions. For example, the volume occupied by an object can be expressed as a product of length, breadth, and height, hence the dimensions of volume are \([L]\times [L]\times [L]=[L]^{3}\text{.}\) Similarly, velocity is a distance divided by a time, and hence the dimensions of velocity are \([L][T]^{-1}\text{.}\) The dimensional equations of speed [v], force [F] and mass density \([\rho]\) can be expressed as \([v]=[M^{o}LT^{-1}]\text{,}\) \([F]=[MLT^{-2}]\text{,}\) and \([\rho]=[ML^{-3}T^{o}]\text{.}\)
Dimensional analysis is a method in which the relationship between different physical quantities (or units) can be analyzed by identifying their base quantities (or units). Dimensional analysis helps convert the units of one system into another system or helps check the resulting formula in a physical process. Some quantities have no dimensions and they are called dimensionless quantities, e.g. the ratio of two lengths are dimensionless. The dimensional consistency does not guarantee the correctness of equations. It is uncertain to the extent of dimensionless quantities or functions. Dimensional analysis is based on the principle of homogeneity which states that the dimensions of both sides of an equation are same. For example, in the equation below dimension on left is equal to dimension on right.
\begin{equation*}
x=x_{o}+v_{o}t+\frac{1}{2}at^{2}.
\end{equation*}
To verify that let us use the principle of homogeneity, i.e.,
\begin{equation*}
[L]=[L]+[LT^{-1}][T] + [LT^{-2}][T]^{2} = [L]+[L]+[L] =3[L]
\end{equation*}
which implies that both sides of the equation has the same dimension, hence this equation is a dimensionally correct. It must be noted that a dimensionally correct equation need not be an exact (correct) equation, but a dimensionally wrong (incorrect) equation must be wrong.
Use principle of homogeneity to find the time period of the oscillation of a simple pendulum which is dependent on its mass, length, and the acceleration due to gravity. From the given information, we have -
\begin{equation*}
time \varpropto mass^a* (length)^b* g^c
\end{equation*}
or,
\begin{equation*}
T = K* M^a* L^b* g^c
\end{equation*}
[where K is a dimensionless quantity]
\begin{equation*}
or,\quad [M^{o}L^{o}T] = [M]^a* [L]^b*[L*T^-2]^c = [M^{a}L^{b+c}T^{-2c}]
\end{equation*}
Equating the power on both sides of the dimensional equation, we get - \(a=0, \quad b+c=0, \quad -2c=1 \Rightarrow \quad c=-\frac{1}{2},\) and \(b=-c=\frac{1}{2},\) \(\qquad\therefore\quad t =K \sqrt{\frac{l}{g}}\)
Some derived units are summarized in the table 1.2.
| Physical Quantity | Formula | Derived SI Units | Symbol |
|---|---|---|---|
| Velocity | \(v/t\) | \(m/s\) | |
| Acceleration | \(v/t^2\) | \(m/s^2\) | |
| Force | \(ma\) | \(kgm/s^2\) | Newton (N) |
| Work (Energy) | \(Fd\) | \(kgm^2/s^2\) | Joule (J) |
| Power | \(W/t\) | \(kgm^2/s^3\) | Watt (W) |
| Pressure | \(Force/A\) | \(kg/ms^{2}\) | Pascal (Pa) |
| Electric Charge | \(It\) | \(As\) | Coulomb (C) |
| Electric Potential | \(IR\) | \(kgm^2/As^3\) | Volt (V) |
| Magnetic Field | \(\mu_{o} I/A\) | \(kg/As^{2}\) | Tesla (T) |
| Magnetic Flux | \(Vt\) | \(kgm^{2}/As^{2}\) | Weber (wb) |
