Units, Physical Quantities, and Dimensional Analysis

Units, Physical Quantities & Dimensional Analysis

Physical Quantities and Units

Physics relies on the measurement of physical quantities, which are properties of objects or phenomena that can be quantified. Each physical quantity is expressed as a numerical value and a unit, which provides a standard for comparison.

• Physical Quantity: Any property that can be measured, such as length, mass, time, etc.

• Unit: A standard measurement used to express a physical quantity (e.g., meter for length, kilogram for mass).

• SI Units: The International System of Units (SI) is the standard system used in physics. The base SI units include meter (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol), and candela (cd).

Example: The speed of a car can be measured as 20 meters per second (20 m/s).

Dimensional Analysis

Dimensional analysis is a method used to check the consistency of equations and to derive relationships between physical quantities. Each physical quantity can be expressed in terms of basic dimensions: length (L), mass (M), time (T), etc.

• Dimension: The nature of a physical quantity, expressed in terms of basic quantities (e.g., [L] for length, [M] for mass, [T] for time).

• Dimensional Formula: An expression showing the powers to which the base quantities are raised in order to represent the quantity (e.g., velocity: [L][T]-1).

Common Dimensional Formulas:

• Velocity: $[L][T]^{-1}$

• Acceleration: $[L][T]^{-2}$

• Force: $[M][L][T]^{-2}$

• Energy: $[M][L]^2[T]^{-2}$

Example: To check if the equation for distance $s = ut + \frac{1}{2}at^2$ is dimensionally correct, substitute the dimensions for each term and verify both sides match.

Derived Units and Conversion

Many physical quantities are derived from the base units. Conversion between units is often necessary in problem-solving.

• Derived Unit: A unit defined by a combination of base units (e.g., newton (N) for force: $1\,N = 1\,kg\cdot m\cdot s^{-2}$).

• Unit Conversion: Changing from one unit to another using conversion factors (e.g., $1\,km = 1000\,m$).

Example: To convert 5 kilometers to meters: $5\,km \times 1000\,m/km = 5000\,m$.

Significant Figures and Scientific Notation

Measurements in physics are reported with significant figures to reflect the precision of the measurement. Scientific notation is used to express very large or very small numbers conveniently.

• Significant Figures: The digits in a measurement that are known with certainty plus one estimated digit.

• Scientific Notation: A way to write numbers as a product of a coefficient and a power of ten (e.g., $3.0 \times 10^8$ m/s).

Example: The speed of light is $3.00 \times 10^8$ m/s (three significant figures).

Summary Table: Dimensions of Common Physical Quantities

Quantity	Symbol	SI Unit	Dimensional Formula
Length	l	meter (m)	$[L]$
Mass	m	kilogram (kg)	$[M]$
Time	t	second (s)	$[T]$
Velocity	v	meter per second (m/s)	$[L][T]^{-1}$
Acceleration	a	meter per second squared (m/s2)	$[L][T]^{-2}$
Force	F	newton (N)	$[M][L][T]^{-2}$