Units & Dimensions : Dimensional Analysis
Dimensions of a Physical Quantity
Review the study guide on units of derived physical quantities where we learned to express the derived quantity in terms of the base quantities. This expression was called the dimensional formula.
The dimensions of a physical quantity are the powers (or exponents) to which the base quantities are raised to represent that quantity.
For example, acceleration can be expressed as
acceleration
= (length)/(time)2
The dimensions of acceleration are
[acceleration] = [LT^{-2}]
Thus, we conclude that
- Acceleration has one dimension in mass, one dimension in length, and –2 dimensions in time. Any quantity raised to zero is equal to 1. So, all those base quantities that do not appear in the dimensional formula of acceleration have dimension 0. For example, here, the dimension of mass is 0.
Dimensional Equations
Here is what Dimensional Equations mean. Pay attention to the words written in bold, which are the key words.
- Equating a physical quantity with its dimensional formula is called the Dimensional Equation of the physical quantity.
- In other words, Dimensional Equations are equations, which represent the dimensions of a physical quantity in terms of the base quantities.
The Dimensional Equation can be obtained from the equation representing the relations between the physical quantities. This is something that we have been doing, writing the quantity on the left hand side of the equation and its dimensions on the right hand side.
Examples -
[velocity] or [v] = [M^{0} L T^{ -1}]
[Force] or [F] = [M L T^{ -2}]
Dimensional Analysis
Dimension of a physical quantity is similar to
- the unit of a physical quantity
- a term in an algebraic equation which is formed by a particular product of the variables, like xy2, x3yz4.
Owing to such a similarity, the golden rule of Dimensional Analysis is:
"Only those physical quantities can be added or subtracted which have the same dimensions(or dimensional formula)."
Thus,
- Checking the Dimensional Consistency of a physical equation means checking that all the terms that are added or subtracted or equated on different sides of the equation must have the same dimensional formula. So, velocity cannot be added to force, or an electric current cannot be subtracted from temperature.
- When magnitudes of two or more physical quantities are multiplied, their units should be treated in the same manner as ordinary algebraic symbols. We can cancel identical units in the numerator and denominator.
- The same is true for dimensions of a physical quantity. We can cancel identical dimensions in the numerator and denominator.
- An equation being dimensionally incorrect means it is an incorrect equation, however, an equation that is dimensionally correct may not necessarily be a correct equation.
Checking the Dimensional Consistency
If the dimensions of all the terms are not same, the equation is wrong. Here is an example of checking the Dimensional Consistency of an equation of motion :
s(displacement) = u(initial velocity) X t(time) + 1/2 X a(acceleration)X t^{2}(time)
Or, s = u.t + 1/2 a.t^{2}
Step 1: Identify the terms in the equation and identify which symbol stands for which physical quantity.
The terms here are : s, u.t, a.t^{2}
Step 2: Write down the dimensional formula of each symbol used in the equation. If you are unclear how to do it, go to the first article in this series.
[s] = [L]
[u] = [LT^{-1}]
[t] = [T]
[a] = [LT^{-2}]
Step 3: Calculate the dimension of each term in the equation.
Term 1: [s] = [L]
Term 2: [u.t] = [LT ^{-1}.T] = [L]
Term 3: [a.t^{2}] = [LT ^{-2}.T^{2}] = [L]
Note that, as stated above, we have canceled dimensions from numerator and denominator like [T ^{-2}.T^{2}] .
Conclusion: The equation is dimensionally consistent since all the terms have the same dimensions.
Note: The following kind of terms are dimensionless in an equation.
- A pure number
- ratio of similar physical quantities, such as angle as the ratio (length/length), refractive index as the ratio (speed of light in vacuum/speed of light in medium) etc.
- The arguments of special functions, such as the trigonometric, logarithmic and exponential functions
Practice Problems
Now that we have dimensional analysis explained, here are some practice problems:
1. [F] = [MLT^{-2}]
What is the dimension of Force in mass? What is the dimension of force in Temperature?
2. Write down the dimensional equation of Density.
3. Check the dimensional consistency of the following equations:
(a) K = (1/2)mv^{2} + ma
(b) K = (1/2)mv^{2}
(c) K = (3/16)mv^{2}
K = Kinetic Energy; m = Mass; v = velocity; a = acceleration
Can you tell on the basis of Dimensional Analysis that which of these is the correct formula for kinetic energy?
[Answer : (a) Dimensionally inconsistent; (b), (c) Dimensionally consistent ; No ]