Dimensions are the physical nature of a quantity and express how it depends on basic physical quantities like mass, length, and time. They are written using symbols: [M] for mass, [L] for length, [T] for time, and so on. Dimensional analysis helps us check the consistency of equations and derive relationships between physical quantities.
There are seven fundamental physical quantities in the SI system. All other physical quantities are derived from these base dimensions:
| Quantity | SI Unit | Dimension Symbol |
|---|---|---|
| Length | meter (m) | [L] |
| Mass | kilogram (kg) | [M] |
| Time | second (s) | [T] |
| Electric Current | ampere (A) | [I] |
| Temperature | kelvin (K) | [Θ] |
| Amount of Substance | mole (mol) | [N] |
| Luminous Intensity | candela (cd) | [J] |
Let’s say a formula is given: \( F = k \cdot \frac{q_1 q_2}{r^2} \), which is Coulomb's Law.
We want to find the dimensions of the constant \( k \).
We know:
Substituting into the equation:
\[ [k] = \frac{[F] \cdot [r]^2}{[q]^2} = \frac{[M L T^{-2}] \cdot [L]^2}{[A T]^2} = \frac{M L^3 T^{-2}}{A^2 T^2} = [M L^3 A^{-2} T^{-4}] \]
Suppose power \( P \) depends on force \( F \) and velocity \( v \). Use dimensional analysis to find the formula.
Let \( P = k \cdot F^a \cdot v^b \)
Dimensions:
\[ [M L^2 T^{-3}] = [M L T^{-2}]^a \cdot [L T^{-1}]^b = M^a L^{a+b} T^{-2a - b} \]
Equating powers:
So the formula is: \( P = k \cdot F \cdot v \)
Written by Matic Zimic