Dimensional analysis is a powerful verification and derivation tool.
Key Dimensions:
| Quantity | Dimension | SI Unit |
|---|---|---|
| Displacement | [L] | m |
| Velocity | [LT^{-1}] | m/s |
| Acceleration | [LT^{-2}] | m/s^2 |
| Time | [T] | s |
Applications:
- Verify equations: All terms must have same dimensions. E.g., v = u + at: [LT^{-1}] = [LT^{-1}] + [LT^{-2}][T] = [LT^{-1}]. Correct.
- Derive relations: If T depends on l and g: [T] = [L]^a[LT^{-2}]^b gives a+b=0, -2b=1, so b=-1/2, a=1/2. T = k*sqrt(l/g).
- Detect errors: If dimensions don't match, the equation is definitely wrong.
Limitation: Cannot determine dimensionless constants (like 2*pi) or distinguish between dimensionally identical quantities.