Physics and Measurement: SI Units, Dimensional Analysis, and Significant Figures

Physics and Measurement

Introduction to Physics and Measurement

Physics is the science that studies the fundamental properties and interactions of matter and energy. It relies on experimental observations and quantitative measurements to describe and predict the behavior of physical systems.

• Measurement is the assignment of a numerical value to a characteristic of an object or event, allowing for comparison with other objects or events.

• Physical properties that can be measured include mass, length, area, volume, and velocity.

• Changes in physical properties are used to describe transformations or evolutions of systems.

• Physical properties are often called observables.

SI Units

International System of Units (SI)

The International System of Units (SI) is the standard framework for measurement in science. It defines seven base quantities, each with a unique unit and symbol.

Base Quantity	Name	Symbol
Length	meter	m
Mass	kilogram	kg
Time	second	s
Electric current	ampere	A
Thermodynamic temperature	kelvin	K
Amount of substance	mole	mol
Luminous intensity	candela	cd

Examples of Measured Quantities

Physical quantities can vary over many orders of magnitude. The following tables provide examples of typical values:

Observable	Length (m)
Distance from Earth to most remote known quasar	$1.4 \times 10^{26}$
Diameter of a hydrogen atom	$1 \times 10^{-10}$
Diameter of a proton	$1 \times 10^{-15}$
Diameter of a person	$1 \times 10^{0}$
Distance from Earth to Sun	$1.5 \times 10^{11}$
Length of a cell of most living organisms	$1 \times 10^{-5}$
Length of a football field	$1 \times 10^{2}$
Distance from Earth to Moon	$3.8 \times 10^{8}$

Observable	Mass (kg)
Universe	$1 \times 10^{53}$
Milky Way galaxy	$1 \times 10^{42}$
Earth	$5.98 \times 10^{24}$
Moon	$7.35 \times 10^{22}$
Electron	$9.11 \times 10^{-31}$

Dimensions and Dimensional Analysis

Physical Dimensions

Dimension denotes the physical nature of a quantity. For example, length can be measured in feet or meters, but both represent the same dimension.

• Symbols for dimensions: L (length), M (mass), T (time).

Dimensions and Units of Derived Quantities

Derived quantities are expressed in terms of base dimensions and units. The following table summarizes some common derived quantities:

Quantity	Dimensions	SI Units	U.S. Units
Area ($A$)	$L^2$	$m^2$	$ft^2$
Volume ($V$)	$L^3$	$m^3$	$ft^3$
Speed ($v$)	$L/T$	$m/s$	$ft/s$
Acceleration ($a$)	$L/T^2$	$m/s^2$	$ft/s^2$

Dimensional Analysis

Dimensional analysis is a method used to check the correctness of equations by ensuring that both sides have the same dimensions.

• Any physical relationship is valid only if the dimensions on both sides of the equation are identical.

• Example: For the equation $x = \frac{1}{2} a t^2$, the dimension form is:

$L = L/T^2 \cdot T^2 = L$

This confirms the equation is dimensionally consistent.

Unit Conversion

Conversion Factors

Conversion between SI and U.S. customary units is often necessary in physics. Common conversion factors for length include:

• 1 mile = 1609 m = 1.609 km

• 1 ft = 0.3048 m = 30.48 cm

• 1 m = 39.37 in = 3.281 ft

• 1 in = 0.0254 m = 2.54 cm

Example: Speed Conversion

Suppose a car travels at 38.0 m/s. Is this faster than a speed limit of 75.0 mi/h?

• Convert meters to miles:

$38.0 \frac{m}{s} \times \frac{1 mi}{1609 m} = 2.36 \times 10^{-2} \frac{mi}{s}$

• Convert seconds to hours:

$2.36 \times 10^{-2} \frac{mi}{s} \times \frac{60 s}{1 min} \times \frac{60 min}{1 h} = 85.0 \frac{mi}{h}$

The driver is exceeding the speed limit.

Order-of-Magnitude

Estimating Orders of Magnitude

An order-of-magnitude estimate expresses a value as a power of ten, providing a rough approximation.

• $0.0086 m \sim 10^{-2} m$

• $0.0021 m \sim 10^{-3} m$

Significant Figures

Accuracy and Uncertainty

Measurements are never perfectly accurate; they are known only within the limits of experimental uncertainty.

• Significant figures (or significant digits) are the digits in a number that contribute to its precision.

Examples:

• $h = 1.6 \pm 0.2 \implies 1.4 \leq h \leq 1.8$

• $1.234 \times 10^{-4} = 0.0001234$ has 4 significant figures

• $1000 = 1 \times 10^3$ has 1 significant figure

• $1001$ has 4 significant figures

Rules for Significant Figures

• Multiplying or dividing: The result should have the same number of significant figures as the quantity with the fewest significant figures.

• Example: $25.57 m \times 2.45 m = 62.6 m^2$ (limited to 3 significant figures by 2.45 m)

• Adding or subtracting: The result should have the same number of decimal places as the term with the fewest decimal places.

• Example: $135 m + 3.25 m = 138 m$ (limited to the units decimal value by 135 m)

Summary Table: Significant Figures Rules

Operation	Rule	Example
Multiplication/Division	Result has same number of significant figures as the least precise factor	$2.45 \times 25.57 = 62.6$
Addition/Subtraction	Result has same number of decimal places as the least precise term	$135 + 3.25 = 138$

Additional info: These notes cover foundational concepts in measurement and units, essential for all physics courses and directly relevant to introductory college physics.