Units & Measurement

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,

Then,

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 = A^p B^q/C^r

Then,ΔZ/Z = p (ΔA/A) + q (ΔB/B) + r (ΔC/C).

Significant Figures:

The significant figures of a number are digits that carry meaningful contributions to its measurement resolution.

All non-zero digits are significant: 1, 2, 3, 4, 5, 6, 7, 8, 9.
Zeros between non-zero digits are significant: 102, 2005, 50009.
Leading zeros are never significant: 0.02, 001.887, 0.000515.
In a number with or without a decimal point, trailing zeros (those to the right of the last non-zero digit) are significant provided they are justified by the precision of their derivation: 389,000; 2.02000; 5.400; 57.5400.

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 × 10^b, 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

In multiplication or division, the final result should retain as many significant figures as are there in the original number with the least significant figures.
In addition or subtraction, the final result should retain as many decimal places as are there in the number with the least decimal places.

Dimensions of Physical Quantities

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]

Checking Dimensional Consistency of an Equation:

Let us consider following equation which gives the distance x travelled by an object in time t. The object starts from position x_{0 with an initial velocity v₀ at time t = 0, and has uniform acceleration a along the direction of motion.}

`x=x_0+v_0t+1/2at^2`

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.