Units, Physical Quantities, and Dimensional Analysis

Units, Physical Quantities & Dimensional Analysis

Introduction to Physical Quantities

Physical quantities are properties of objects or phenomena that can be measured. Each physical quantity is expressed as the product of a numerical value and a unit. Understanding the correct use of units and dimensional analysis is fundamental in Physics with Calculus.

• Physical Quantity: Any property that can be quantified by measurement (e.g., length, mass, time).

• Unit: A standard reference used to express the measurement of a physical quantity (e.g., meter, kilogram, second).

• Measurement: The process of determining the value of a physical quantity using a standard unit.

Fundamental and Derived Quantities

Physical quantities are classified as either fundamental (base) or derived. Fundamental quantities are independent and form the basis for all other measurements, while derived quantities are combinations of fundamental quantities.

• Fundamental Quantities: Length (L), Mass (M), Time (T), Electric Current (I), Temperature (Θ), Amount of Substance (N), Luminous Intensity (J).

• Derived Quantities: Quantities that are combinations of fundamental quantities (e.g., velocity, acceleration, force).

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 its fundamental dimensions.

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

• Example: The dimension of velocity is $[L][T]^{-1}$, since velocity is defined as displacement divided by time.

Common Derived Quantities and Their Dimensions

Quantity	Symbol	Unit	Dimension
Velocity	v	m/s	$[L][T]^{-1}$
Acceleration	a	m/s2	$[L][T]^{-2}$
Force	F	Newton (N)	$[M][L][T]^{-2}$
Energy	E	Joule (J)	$[M][L]^2[T]^{-2}$

Dimensional Homogeneity

All terms in a physically meaningful equation must have the same dimensions. This principle is called dimensional homogeneity and is used to verify the correctness of equations.

• Example: In the equation for displacement under constant acceleration, $s = ut + \frac{1}{2}at^2$, each term has the dimension of length $[L]$.

Applications of Dimensional Analysis

• Checking the correctness of equations.

• Deriving relationships between physical quantities.

• Converting units from one system to another.

Example Problem

Problem: Find the dimensions of the gravitational constant $G$ in Newton's law of gravitation: $F = G \frac{m_1 m_2}{r^2}$.

• Solution: Rearranging for $G$ gives $G = \frac{F r^2}{m_1 m_2}$.

• Substitute dimensions: $[F] = [M][L][T]^{-2}$, $[r] = [L]$, $[m_1] = [M]$, $[m_2] = [M]$.

• So, $[G] = \frac{[M][L][T]^{-2} [L]^2}{[M][M]} = [M]^{-1}[L]^3[T]^{-2}$.