Dimensional Analysis CUED 1A Lecture notes 2 Units 4 Dimensionless forms of equations

Chapter 3 Dimensions

Units are defined by humans whereas dimensions arise from nature. The property of a quantity to be represented by a set of inter-convertible units is termed as its dimension. The radius of the earth is 6400 km = 3976 miles = 31814 furlongs = $7\times 10^{6}$ yards, while the mass of earth is $5.9\times 10^{24}$ kg = $1.3\times 10^{25}$ lbm. The independence of the units of these two quantities is encapsulated by assigning them a dimension. The earth’s radius has dimensions of length and the mass has dimensions of mass. All units of length form a so-called equivalence class (that is the technical term) by virtue of being convertible between themselves and so do the units of mass. The concept of dimensions thus removes the human-dependent definition of the system of units and abstracts the equivalence-independence relationship between units of physical quantities.

3.1 Notation

Dimensions are symbolically denoted by a letter, as shown in table 3.1.

Quantity	SI unit	Dimensions
Mass	kg	$M$
Length	m	$L$
Time	s	$T$
Electric current	A	$I$
Temperature	K	$\Theta$

Table 3.1: Base dimensions in the (nominal) MLT system, along with their SI units.

Advanced tip.

The concept and notation for dimensions was first explicated by James Clerk Maxwell in 1871, who was then the Cavendish Professor of Physics at Cambridge. Maxwell enclosed the symbol for dimension within square brackets, so that $[L]$ denoted the dimension of length instead of simply $L$ (this convention is followed by many to date). The reason was to not conflate the dimension $[L]$ with any symbol $L$ that perhaps denoted some length. This notation with the square brackets is still used today in many sources to denote dimensions.

3.2 Base and derived dimensions

The property of base and derived units carries over to dimensions. The symbols in table 3.1 denote the MLT system, where mass, length and time form the base dimensions. Dimensions of derived quantities are determined in the same way as for derived units.

Question 3.1.

Determine the dimensions of velocity, acceleration, force in the MLT system of dimensions.

Answer 3.1.

$L/T$ , $L/T^{2}$ and $ML/T^{2}$ , respectively (see table 3.2 for more).

Quantity		Dimensions
Velocity	length/time	$L/T$
Acceleration	rate of change of velocity	$L/T^{2}$
Force	mass times acceleration	$ML/T^{2}$
Density	mass per unit volume	$M/L^{3}$
Momentum	mass times velocity	$ML/T$
Work	Force times distance	$ML^{2}/T^{2}$
Power	Rate of work	$ML^{2}/T^{3}$
Pressure	Force per unit area	$M/(LT^{2})$

Table 3.2: Some derived dimensions in the MLT system.

3.3 System of dimensions

Systems of dimensions are distinguished by the choice of base dimensions. Just as the $M L T$ system, one may define the $F L T$ system to mirror the MKS force system of units, where the base dimensions are force ( $F$ ), length ( $L$ ) and time ( $T$ ). In such a system, the derived quantities are derived in table 3.3.

Quantity		Dimensions
Velocity	length/time	$L/T$
Acceleration	rate of change of velocity	$L/T^{2}$
Mass	force divided by acceleration	$FT^{2}/L$
Density	mass per unit volume	$FT^{2}/L^{4}$
Momentum	mass times velocity	$F T$
Work	Force times distance	$F L$
Power	Rate of work	$FL/T$
Pressure	Force per unit area	$F/L^{2}$

Table 3.3: Some derived dimensions in the FLT system.

3.4 Specification of dimensional quantities

(from Dr John P Longley’s Lecture Notes)
To completely specify any quantity that has dimensions it is necessary to give two pieces of information, the numerical value and the units in which it is measured, e.g.,

\displaystyle\text{time:}\qquad t=2.5\text{ s},\qquad\text{or}\qquad\text{% distance:}\qquad d=35.4\text{ km}.

(3.1)

A single space is always left between the value and the unit, and neither should be italicized.

3.5 Principle of dimensional consistency

This principle is the foundation upon which the structure of dimensional analysis is erected.

Principle 3.1.

There are two parts:

1.

The dimension of product (or ratio) of two quantities is the product (or ratio) of the dimensions.
2.

Only quantities with identical dimensions may be added, subtracted, equated or compared with each other. This principle is also called the principle of dimensional homogeneity.

Note that we have been following principle 3.1-1 in constructing the derived dimensions. And we tacitly used principle 3.1-2 in Questions 1.1-1.4

Consider the following two examples as demonstration.

Question 3.2.

(from Dr John P Longley’s Lecture Notes)
Show that the equation for the period, $t$ , of a simple pendulum as it depends on the length of the pendulum, $l$ , and the acceleration due to gravity, $g$ ,

\displaystyle t=2\pi\sqrt{\dfrac{l}{g}}

(3.2)

is dimensionally consistent.

Answer 3.2.

The dimensions of $t$ are $T$ , of $l$ are $L$ and of $g$ are $L/T^{2}$ . Therefore, the dimensions of the two sides of the equation are

\displaystyle\text{LHS}=T,\qquad\text{RHS}=\sqrt{\dfrac{L}{L/T^{2}}}=T.

(3.3)

In determining the dimensions of the RHS, we employed principle 3.1-1 of dimensional consistency, and the dimensions of two sides being equal demonstrates principle 3.1-2.

Question 3.3.

Show that the expression (1.3) for the natural frequency of oscillation of a stretched string is dimensionally consistent. For convenience, here is the equation again

\displaystyle f=\dfrac{1}{2}\sqrt{\dfrac{\tau}{ml}},

where $f$ is the frequency, $\tau$ is the tension in the string, $m$ is its mass and $l$ its length.

Answer 3.3.

The dimension of the LHS (frequency) are $1/T$ . The dimensions of the tension $\tau$ (which is a force) are $ML/T^{2}$ , while those of $m$ are $M$ and $l$ are $L$ . The dimensions of the RHS are to be determined according to principle 3.1-1 as

\displaystyle RHS=\sqrt{\dfrac{ML/T^{2}}{ML}}=\dfrac{1}{T}\quad.

Since both the dimensions of the LHS and RHS are $1/T$ , expression (1.3) obeys principle 3.1-2 of dimensional consistency.

It is also useful to examine an example that violates dimensional consistency.

Question 3.4.

In cardiology, the peak pressure difference, $\Delta p$ , between the left ventricle and the aorta, measured in mm of Hg, is determined from the peak speed, $v$ , of blood at the inlet of the aorta, measured in m/s, using the expression

\displaystyle\Delta p=4v^{2}.

Is this expression dimensionally consistent?

Answer 3.4.

The dimensions of the LHS are $M/(LT^{2})$ , whereas those of the RHS (determined using principle 3.1-1 of dimensional consistency) are $L^{2}/T^{2}$ . The two sides have different dimensions and, therefore, violate principle 3.1-2 of dimensional consistency.

Now we come to the crux of the matter. Dimensional consistency implies that the equation looks the same in any consistent set of units.

Principle 3.2.

Dimensional consistency: A complete statement of a physical law is independent of the system of measurement.

In other words, the same expression relating multiple physical quantities holds in any consistent set of units. We state that the Principle 3.2 implies and is implied by Principle 3.1. (Interested reader may find the proof in “Dimensional Analysis and Theory of Models” by Henry L Langhar, John Wiley & Sons, Inc., New York, 1951.) Let us demonstrate this using equation (3.2) from Question 3.2, the example of the simple pendulum.

Question 3.5.

(from Dr John P Longley’s Lecture Notes)
Assume that the equation (3.2) holds in the SI system of units. Define a custom set of units using 1 arm-length = 0.8 m and 1 heart-beat = (2/3) s. Show that (3.2) also holds in these new sets of units.

Answer 3.5.

Let us abbreviate arm-length to al and heart-beat to hb. So 1 al = 0.8 m and 1 hb = (2/3) s. The numerical value of $t$ , $g$ and $l$ in the two sets of units will be different (owing to the conversion factors), and so we will denote those differently. Let $t_{\text{value}}$ , $l_{\text{value}}$ and $g_{\text{value}}$ be the numerical values of $t$ , $l$ and $g$ , respectively, in SI units. And let $t^{\prime}_{\text{value}}$ , $l^{\prime}_{\text{value}}$ and $g^{\prime}_{\text{value}}$ be them in the new units. Applying the conversion factors,

$\displaystyle t^{\prime}_{\text{value}}$	$\displaystyle=t_{\text{value}}\times\dfrac{1\text{ hb}}{(2/3)\text{ s}}$	$\displaystyle=$	$\displaystyle\dfrac{3}{2}~{}t_{\text{value}},$
$\displaystyle l^{\prime}_{\text{value}}$	$\displaystyle=l_{\text{value}}\times\dfrac{1\text{ al}}{0.8s}$	$\displaystyle=$	$\displaystyle\dfrac{5}{4}~{}l_{\text{value}},$
$\displaystyle g^{\prime}_{\text{value}}$	$\displaystyle=g_{\text{value}}\times\dfrac{(2/3)^{2}\text{ s}^{2}}{1\text{ hb}% ^{2}}\times\dfrac{1\text{ al}}{0.8\text{ m}}$	$\displaystyle=$	$\displaystyle\dfrac{5}{9}~{}g_{\text{value}},$

Now notice how the expression $\sqrt{l/g}$ transforms

\displaystyle\sqrt{\dfrac{l^{\prime}_{\text{value}}}{g^{\prime}_{\text{value}}% }}=\sqrt{\dfrac{(5/4)~{}l_{\text{value}}}{(5/9)~{}g_{\text{value}}}}=\dfrac{3}% {2}~{}\sqrt{\dfrac{l_{\text{value}}}{g_{\text{value}}}}~{}

exactly as $t$ does (by a factor of (3/2)). Therefore, both

\displaystyle t_{\text{value}}=2\pi\sqrt{\dfrac{l_{\text{value}}}{g_{\text{% value}}}}\qquad\text{and}\qquad t^{\prime}_{\text{value}}=2\pi\sqrt{\dfrac{l^{% \prime}_{\text{value}}}{g^{\prime}_{\text{value}}}}

(3.4)

hold. In general, for any simple pendulum that obeys equation (3.2), the values of the parameters can be substituted in any consistent system of units.

Question 3.6.

Let us revisit the expression in question 3.3 for the expression for the natural frequency of a stretched string. Show that converting from SI to custom units for length (length-units, abbreviated lu), mass (mass units, abbreviated mu) and time (time-units, abbreviated tu) with (symbolic) conversion factors $\lambda$ , $\mu$ and $\tau$ defined as

\displaystyle 1\text{ m}=\lambda\text{ lu},\qquad 1\text{ kg}=\mu\text{ mu},% \qquad\text{and}\qquad 1\text{ s}=\tau\text{ tu},

leaves the expression invariant.

Answer 3.6.

Left as an exercise for the reader.

Why does dimensional consistency imply that an equation is independent of system of units? Without loss of generality, let us consider an equation described by the MLT set of base dimensions. The principle 3.1-2 of dimensional consistency implies that each term on the left and right hand side of the equation (or inequality) has the same dimensions, say $M^{a}L^{b}T^{c}$ for some exponents $a$ , $b$ and $c$ . If $\mu$ , $\lambda$ and $\tau$ are the conversion factors between two sets of units for mass, length and time respectively (just as in question 3.6), then each term in the equation will accumulate the same conversion factor of $\mu^{a}\lambda^{b}\tau^{c}$ . (This factor follows from principle 3.1-1 of dimensional consistency. By definition, the conversion factors are non-zero.) Therefore, canceling this factor from each term recovers the same equation in the new system of units.

3.6 Dimensionless combinations of parameters

A combination of parameters that end up with dimensions of unity is called a dimensionless parameter, dimensionless group or dimensionless number. An excellent example of a dimensionless number is the Mach number. For an aircraft flying with speed $v$ in a medium with the speed of sound $c$ , its Mach number is the ratio $v/c$ . Each of $v$ and $c$ are speeds, so have dimensions $L/T$ . But the Mach number is a ratio of the two and therefore has dimensions of unity (in other words $M^{0}L^{0}T^{0}$ ).

Here are a few other examples.

Question 3.7.

Show that the coefficient of static friction is dimensionless.

Answer 3.7.

The coefficient of static friction ( $\mu$ ) is used to determine the maximum force of friction between static surfaces in contact using the equation

\displaystyle f=\mu n,

(3.5)

where $f$ is the magnitude of the strongest force of friction and $n$ is the normal force of compression between the two surfaces. Naturally, $\mu$ has the dimensions of the ratio of two forces, and is, therefore, dimensionless.

Ratio of quantities with identical dimensions is not the only way to get dimensionless numbers. Consider the following question.

Question 3.8.

The coefficient of lift, $C_{L}$ , of a wing generating a lift force (e.g. an airplane wing or a helicopter blade) is defined as the ratio

\displaystyle C_{L}=\dfrac{l}{\dfrac{1}{2}\rho v^{2}a}

(3.6)

where $\ell$ is the lift force generated by the wing, $\rho$ is the density of the medium, $v$ is the speed of the wing relative to the medium and $a$ is the area of the wing. Show that $C_{L}$ is dimensionless. An F22 raptor weighs 30000 kg and has a wing area of 78 m². When cruising at a speed of 1482 km/hr at sea level, what is the lift coefficient?

Answer 3.8.

The lift force in the numerator has dimensions of $ML/T^{2}$ . The denominator derives it dimensions from $\rho v^{2}a$ , which are

\displaystyle\dfrac{M}{L^{3}}\times\left(\dfrac{L}{T}\right)^{2}\times L^{2}=% \dfrac{ML}{T^{2}}.

The numerator and denominator both have dimensions of force, so their ratio is dimensionless.

For the F22 raptor (at sea level where the air density is 1.2 kg/m³), the lift coefficient is

\displaystyle C_{L}=\dfrac{30000\text{ kg}\times 9.8\text{ m/s${}^{2}$}}{% \dfrac{1}{2}\times 1.2\text{ kg/m${}^{3}$}\times\left(1482\dfrac{\text{ km}}{% \text{ hr}}\times\dfrac{1000\text{ m}}{1\text{ km}}\times\dfrac{1\text{ hr}}{3% 600\text{ s}}\right)^{2}\times 78\text{ m${}^{2}$}}=0.0371.

(3.7)

Although the lift coefficient is not technically ratio of two forces, it may be written as the ratio of two quantities that have dimensions of force. A moment’s thought will convince you that a dimensionless number can be expressed as a ratio of two forces, two lengths, two velocities, two times, or any two quantities whose dimensions can be constructed from the constituent dimensions of the parameters making up the dimensionless number. A conveniently constructed such ratio is usually used as an interpretation of the dimensionless number.

Question 3.9.

(from Dr John P Longley’s Lecture Notes)
Calculate the number of car-lengths travelled in a half-hour long car journey at 35 miles/hr. Assume the car is 14 ft long.

Answer 3.9.

\displaystyle\text{Ratio}=\dfrac{35\dfrac{\text{ miles}}{\text{ hr}}\times 0.5% \text{ hr}}{14\text{ ft}\times\dfrac{1\text{ mile}}{5280\text{ ft}}}=6.60% \times 10^{3}.

This number will be the same in any system of units because it is dimensionless.

Other examples of dimensionless numbers are constants ( $\pi$ , 17.3, $e$ , etc), arguments and values of mathematical functions (e.g. both $\theta$ and $\sin\theta=\theta-\theta^{2}/3!+\theta^{5}/5!\dots$ ).

Principle 3.3.

A property of dimensionless numbers is that they have the same value in any (consistent) set of units, i.e. the conversion factor for these parameters between systems of units is unity (in the notation of Question 3.6, the factor is $\mu^{0}\lambda^{0}\tau^{0}=1$ ).

3.7 Conclusion

Dimensions abstract that property of quantities which enable them to be expressed in a set of inter-convertible units. When a physical law is independent of the system of units when expressed in a dimensionally consistent form. This implies that any complete statement of a physical law can be stated in terms of dimensionless variable alone. We will see in Chapter 5 the central role Dimensionless numbers play in dimensional analysis.

Can now do Examples Paper Q1-4.