Dimensions of Physical Quantities
When we talk about the physical dimension of a quantity, we're referring to the fundamental nature of that quantity, irrespective of the measurement units used to express it.
This concept lets us classify and analyze physical quantities by their essence, without needing to consider specific units.
For example, both the distance between two cities and the thickness of a sheet of paper share the same physical dimension: length, which is represented by the symbol \([L]\). Similarly, whether the radius of a galaxy is measured in kilometers or light-years, it’s still a measure of length.
Fundamental quantities stand out as the building blocks of all other measurements, each with its own physical dimension denoted by a unique symbol:
- Time \([T]\)
- Length \([L]\)
- Mass \([M]\)
If \( x \) represents a distance, then its physical dimension will be \([x] = [L]\). If \( t \) represents time, we write \([t] = [T]\).
These fundamental quantities can combine to describe derived quantities by using their respective physical dimensions.
For example, volume has the dimensions of length cubed, written as \([L]^3\). Density, which is mass divided by volume, has dimensions \([M]/[L]^3\). Velocity, a measure of length per unit of time, is represented as \([L]/[T]\). Energy has more complex dimensions, resulting from mass times length squared divided by time squared, or \([M][L]^2/[T]^2\).
To determine the physical dimensions of a derived quantity, we express it in terms of the fundamental quantities, disregarding numerical coefficients as they don’t impact the dimensions.
Here are some examples of derived physical dimensions based on length:
| Quantity | Dimension |
|---|---|
| Distance | \([L]\) |
| Area | \([L]^2\) |
| Volume | \([L]^3\) |
| Density | \([M]/[L]^3\) |
| Velocity | \([L]/[T]\) |
| Acceleration | \([L]/[T]^2\) |
| Energy | \([M][L]^2/[T]^2\) |
Dimensional Consistency in Physical Laws
A physical law is dimensionally consistent if both sides of the equation have the same dimensions.
This principle ensures the validity of physical formulas and the mathematical operations performed on quantities.
The golden rule is that only quantities with the same physical dimension can be added or subtracted.
For instance, let’s say you have a distance of \(5 \, \text{meters}\) and a time of \(3 \, \text{seconds}\). Since length has the physical dimension \([L]\) and time has \([T]\), you can’t add them directly because they’re fundamentally different types of quantities. Adding them would yield no meaningful result. It’s like saying “5 meters plus 3 seconds”—it doesn’t make sense because they don’t share the same physical dimension. On the other hand, if you add two lengths, like \(5 \, \text{meters}\) and \(3 \, \text{meters}\), both with the physical dimension \([L]\), the addition is dimensionally valid, resulting in \(5 + 3 = 8 \, \text{meters}\). Similarly, adding two time durations, like \(2 \, \text{seconds}\) and \(3 \, \text{seconds}\), both with dimension \([T]\), gives \(5 \, \text{seconds}\), a dimensionally consistent outcome.
Therefore, understanding the physical dimensions of quantities allows us not only to classify and distinguish between types of measurements but also to verify the dimensional consistency of physical formulas through dimensional analysis.
Every correct calculation must maintain balance in physical dimensions.
This principle isn’t just a matter of rigor; it’s an essential guide to preventing errors and misunderstandings in the physical sciences.
An Example
Consider the formula for average velocity:
$$ v = \frac{d}{t} $$
where \( v \) stands for velocity, \( d \) for distance, and \( t \) for time.
To check the dimensional consistency of this formula, we replace each variable with the dimension of its respective quantity:
$$ [v] = \frac{[L]}{[T]} $$
This shows that on the left, we have the dimension of velocity, \( [v] = [L]/[T] \), and on the right, we get the same dimension ratio, \( [L]/[T] \), derived from distance divided by time.
Since the dimensions match on both sides of the equation, we can conclude that the formula is dimensionally correct.
