Error in case of a measured quantity raised to a power: The relative error in a physical quantity raised to the power k is the k times the relative error in the individual quantity.
Suppose Z = A2,
ΔZ/Z = (ΔA/A) + (ΔA/A) = 2 (ΔA/A).
Hence, the relative error in A2 is two times the error in A.
In general, if Z = Ap Bq/Cr
Then,ΔZ/Z = p (ΔA/A) + q (ΔB/B) + r (ΔC/C).
The significant figures of a number are digits that carry meaningful contributions to its measurement resolution.
Scientific Notation: To remove ambiguities in determining the number of significant figures, the best way is to report every measurement in scientific notation (in the power of 10). In this notation, every number is expressed as a × 10b, where a is a number between 1 and 10, and b is any positive or negative exponent of 10.
Rules for Arithmetic Operations with Significant Figures
The dimensions of a physical quantity are the powers (or exponents) to which the base quantities are raised to represent that quantity.
In mechanics, all the physical quantities can be written in terms of the dimensions [L], [M] and [T].
Example: Volume = [L] × [L] × [L] = [L3]
Force = Mass × Acceleration
= mass × length/time2
= [M] × [L]/[T2]
= [M] × [L] × [T-2]
Let us consider following equation which gives the distance x travelled by an object in time t. The object starts from position x0 with an initial velocity v0 at time t = 0, and has uniform acceleration a along the direction of motion.
The dimensions of each term may be written as
[x] = [L]
[x0 ] = [L]
[v0 t] = [L T–1] [T] = [L]
[(1/2) a t2] = [L T–2] [T2] = [L]
As each item on RHS of this equation has same dimension as that on LHS of equation, the equation is dimensionally consistent.
A dimensionally correct equation need not be actually an exact (correct) equation, but a dimensionally wrong (incorrect) or inconsistent equation must be wrong.
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