# 1.4 Dimensional analysis  (Page 3/6)

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## Checking equations for dimensional consistency

Consider the physical quantities $s,$ $v,$ $a,$ and $t$ with dimensions $\left[s\right]=\text{L},$ $\left[v\right]={\text{LT}}^{-1},$ $\left[a\right]={\text{LT}}^{-2},$ and $\left[t\right]=\text{T}.$ Determine whether each of the following equations is dimensionally consistent: (a) $s=vt+0.5a{t}^{2};$ (b) $s=v{t}^{2}+0.5at;$ and (c) $v=\text{sin}\left(a{t}^{2}\text{/}s\right).$

## Strategy

By the definition of dimensional consistency, we need to check that each term in a given equation has the same dimensions as the other terms in that equation and that the arguments of any standard mathematical functions are dimensionless.

## Solution

1. There are no trigonometric, logarithmic, or exponential functions to worry about in this equation, so we need only look at the dimensions of each term appearing in the equation. There are three terms, one in the left expression and two in the expression on the right, so we look at each in turn:
$\begin{array}{c}\left[s\right]=\text{L}\hfill \\ \left[vt\right]=\left[v\right]·\left[t\right]={\text{LT}}^{-1}·\text{T}={\text{LT}}^{0}=\text{L}\hfill \\ \left[0.5a{t}^{2}\right]=\left[a\right]·{\left[t\right]}^{2}={\text{LT}}^{-2}·{\text{T}}^{2}={\text{LT}}^{0}=\text{L}\text{.}\hfill \end{array}$

All three terms have the same dimension, so this equation is dimensionally consistent.
2. Again, there are no trigonometric, exponential, or logarithmic functions, so we only need to look at the dimensions of each of the three terms appearing in the equation:
$\begin{array}{c}\left[s\right]=\text{L}\hfill \\ \left[v{t}^{2}\right]=\left[v\right]·{\left[t\right]}^{2}={\text{LT}}^{-1}·{\text{T}}^{2}=\text{LT}\hfill \\ \left[at\right]=\left[a\right]·\left[t\right]={\text{LT}}^{-2}·\text{T}={\text{LT}}^{-1}.\hfill \end{array}$

None of the three terms has the same dimension as any other, so this is about as far from being dimensionally consistent as you can get. The technical term for an equation like this is nonsense .
3. This equation has a trigonometric function in it, so first we should check that the argument of the sine function is dimensionless:
$\begin{array}{c}\left[\frac{a{t}^{2}}{s}\right]=\frac{\left[a\right]·{\left[t\right]}^{2}}{\left[s\right]}=\frac{{\text{LT}}^{-2}·{\text{T}}^{2}}{\text{L}}=\frac{\text{L}}{\text{L}}=1.\hfill \end{array}$

The argument is dimensionless. So far, so good. Now we need to check the dimensions of each of the two terms (that is, the left expression and the right expression) in the equation:
$\begin{array}{c}\left[v\right]={\text{LT}}^{-1}\hfill \\ \left[\text{sin}\left(\frac{a{t}^{2}}{s}\right)\right]=1.\hfill \end{array}$

The two terms have different dimensions—meaning, the equation is not dimensionally consistent. This equation is another example of “nonsense.”

## Significance

If we are trusting people, these types of dimensional checks might seem unnecessary. But, rest assured, any textbook on a quantitative subject such as physics (including this one) almost certainly contains some equations with typos. Checking equations routinely by dimensional analysis save us the embarrassment of using an incorrect equation. Also, checking the dimensions of an equation we obtain through algebraic manipulation is a great way to make sure we did not make a mistake (or to spot a mistake, if we made one).

Check Your Understanding Is the equation v = at dimensionally consistent?

yes

One further point that needs to be mentioned is the effect of the operations of calculus on dimensions. We have seen that dimensions obey the rules of algebra, just like units, but what happens when we take the derivative of one physical quantity with respect to another or integrate a physical quantity over another? The derivative of a function is just the slope of the line tangent to its graph and slopes are ratios, so for physical quantities v and t , we have that the dimension of the derivative of v with respect to t is just the ratio of the dimension of v over that of t :

$\left[\frac{dv}{dt}\right]=\frac{\left[v\right]}{\left[t\right]}.$

Similarly, since integrals are just sums of products, the dimension of the integral of v with respect to t is simply the dimension of v times the dimension of t :

$\left[\int vdt\right]=\left[v\right]·\left[t\right].$

By the same reasoning, analogous rules hold for the units of physical quantities derived from other quantities by integration or differentiation.

## Summary

• The dimension of a physical quantity is just an expression of the base quantities from which it is derived.
• All equations expressing physical laws or principles must be dimensionally consistent. This fact can be used as an aid in remembering physical laws, as a way to check whether claimed relationships between physical quantities are possible, and even to derive new physical laws.

## Problems

A student is trying to remember some formulas from geometry. In what follows, assume $A$ is area, $V$ is volume, and all other variables are lengths. Determine which formulas are dimensionally consistent. (a) $V=\pi {r}^{2}h;$ (b) $A=2\pi {r}^{2}+2\pi rh;$ (c) $V=0.5bh;$ (d) $V=\pi {d}^{2};$ (e) $V=\pi {d}^{3}\text{/}6.$

Consider the physical quantities s , v, a, and t with dimensions $\left[s\right]=\text{L},$ $\left[v\right]={\text{LT}}^{-1},$ $\left[a\right]={\text{LT}}^{-2},$ and $\left[t\right]=\text{T}.$ Determine whether each of the following equations is dimensionally consistent. (a) ${v}^{2}=2as;$ (b) $s=v{t}^{2}+0.5a{t}^{2};$ (c) $v=s\text{/}t;$ (d) $a=v\text{/}t.$

a. Yes, both terms have dimension L 2 T -2 b. No. c. Yes, both terms have dimension LT -1 d. Yes, both terms have dimension LT -2

Consider the physical quantities $m,$ $s,$ $v,$ $a,$ and $t$ with dimensions [ m ] = M, [ s ] = L, [ v ] = LT –1 , [ a ] = LT –2 , and [ t ] = T. Assuming each of the following equations is dimensionally consistent, find the dimension of the quantity on the left-hand side of the equation: (a) F = ma ; (b) K = 0.5 mv 2 ; (c) p = mv ; (d) W = mas ; (e) L = mvr .

Suppose quantity $s$ is a length and quantity $t$ is a time. Suppose the quantities $v$ and $a$ are defined by v = ds / dt and a = dv / dt . (a) What is the dimension of v ? (b) What is the dimension of the quantity a ? What are the dimensions of (c) $\int vdt,$ (d) $\int adt,$ and (e) da / dt ?

a. [v] = LT –1 ; b. [a] = LT –2 ; c. $\left[\int vdt\right]=\text{L;}$ d. $\left[\int adt\right]={\text{LT}}^{–1}\text{;}$ e. $\left[\frac{da}{dt}\right]={\text{LT}}^{–3}$

Suppose [V] = L 3 , $\left[\rho \right]={\text{ML}}^{–3},$ and [t] = T. (a) What is the dimension of $\int \rho dV?$ (b) What is the dimension of dV / dt ? (c) What is the dimension of $\rho \left(dV\text{/}dt\right)?$

The arc length formula says the length $s$ of arc subtended by angle $Ɵ$ in a circle of radius $r$ is given by the equation $s=rƟ.$ What are the dimensions of (a) s , (b) r , and (c) $\text{Ɵ?}$

a. L; b. L; c. L 0 = 1 (that is, it is dimensionless)

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