Every physical quantity can be written in terms of seven SI base quantities; dimensions let you check equations and convert units, and error analysis tells you how reliable a measured result is.
How it's asked: Dimensional formula of a derived quantity (Planck's constant, G, viscosity, surface tension), checking which equation is dimensionally correct, percentage error in a product/power formula, and counting significant figures.
Why this is true
A physical equation equates quantities of the same kind, so every term on both sides must carry the same powers of M, L and T — this is the principle of homogeneity. That is why you can reject a wrong formula without knowing the physics behind it, and why two quantities with the same dimensions (work and torque, or Planck's constant and angular momentum) can still be different physical ideas. Errors combine because a result is computed from several measured quantities: in a sum or difference the absolute errors add, while in a product or quotient the fractional errors add. A quantity raised to the power n contributes n times its fractional error, so the quantity with the highest power dominates the error budget. Dimensional analysis cannot find pure numbers like 2π, and cannot handle trigonometric, exponential or logarithmic terms, because these are dimensionless.
Key points
Key formulae
Force
[F] = [MLT⁻²]
newton = kg m s⁻²
Error in a product/power
ΔZ/Z = pΔA/A + qΔB/B + rΔC/C
for Z = AᵖBᵠ/Cʳ; errors always add
Error in a sum or difference
ΔZ = ΔA + ΔB
absolute errors add
Vernier callipers
LC = 1 MSD − 1 VSD
Screw gauge
LC = pitch / number of circular divisions
Dimensional formulae worth memorising
| Quantity | Defining relation | Dimensions |
|---|---|---|
| Power | work / time | [ML²T⁻³] |
| Pressure, stress, Young's modulus | force / area | [ML⁻¹T⁻²] |
| Planck's constant h | E / ν | [ML²T⁻¹] |
| Gravitational constant G | Fr² / m₁m₂ | [M⁻¹L³T⁻²] |
| Surface tension | force / length | [MT⁻²] |
| Coefficient of viscosity η | F / (A · dv/dx) | [ML⁻¹T⁻¹] |
| Impulse, momentum | F × t | [MLT⁻¹] |
Common traps
- Errors never subtract: for Z = A/B or Z = A − B the error terms are still added.
- A dimensionally correct equation need not be physically correct — the check only rules equations out.
Worked example
g is found from a simple pendulum using g = 4π²L/T². L is measured with a 1% error and T with a 2% error. What is the maximum percentage error in g?
High-yield
Fractional errors add, weighted by the power: the quantity with the largest exponent dominates the percentage error.
Quick check
Q1.Which pair of quantities has the same dimensions?
Q2.The number of significant figures in 0.003050 is: