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 by comparison with a 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 as expressed in terms of fundamental quantities (e.g., [L] for length, [M] for mass, [T] for time).

• Dimensional Formula: An expression showing how a derived quantity depends on fundamental quantities (e.g., velocity: $[L][T]^{-1}$).

Examples of Dimensional Formulas

• Velocity: $[L][T]^{-1}$

• Acceleration: $[L][T]^{-2}$

• Force: $[M][L][T]^{-2}$

• Energy: $[M][L]^2[T]^{-2}$

SI Units

The International System of Units (SI) is the standard system used in physics. Each fundamental quantity has a corresponding SI unit.

Quantity	Symbol	SI Unit	Unit Symbol
Length	L	meter	m
Mass	M	kilogram	kg
Time	T	second	s
Electric Current	I	ampere	A
Temperature	Θ	kelvin	K
Amount of Substance	N	mole	mol
Luminous Intensity	J	candela	cd

Dimensional Consistency and Checking Equations

Equations in physics must be dimensionally consistent; both sides of an equation must have the same dimensions. This principle is used to verify the correctness of equations and to derive relationships between physical quantities.

• Example: For the equation $s = ut + \frac{1}{2}at^2$, the dimensions of each term are $[L]$ (length), confirming dimensional consistency.

Applications of Dimensional Analysis

• Checking the validity of equations

• Deriving formulas when the relationship between quantities is unknown

• Converting units from one system to another