Propagation of Error Power

Description:

When measurements are raised to a power, the fractional error in a physical quantity raised to the power k is the k times the fractional error in the individual quantity.

Consider a measurement – a with error Δa, that is to be raised to a power b.

Let C be the resulting measurement with error ΔC

C = (a ± Δa)^b

ΔC/C = b(Δa/a)

∴ ΔC = b(Δa/a)C

Example 1

Radius is measured as 5 ± 1

Then the area of circle is given by −

π(5 ± 1)²

A = 25π

ΔA/A = ±2 ×1/5 = 2/5

∴ ΔA = 2/5 × A = 2/5 = 25 π = 10 π

Hence,

Area = 25π ± 10 π

Or

Area = 25π ± 40%