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 of matter, energy, space, and time. It relies on experimental observations and quantitative measurements to describe and understand natural phenomena.

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

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

• Changes in physical properties describe transformations or evolutions of a system.

• 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 Lengths and Masses

Physical quantities can span many orders of magnitude. Below are examples of measured lengths and masses:

Observable	Length (m)
Distance from Earth to nearest known quasar	$1.4 \times 10^{26}$
Diameter of a proton	$1 \times 10^{-15}$
Diameter of a person	$1 \times 10^{0}$

Observable	Mass (kg)
Universe	$1 \times 10^{53}$
Earth	$5.98 \times 10^{24}$
Electron	$9.11 \times 10^{-31}$

*Additional info: Only selected entries shown for brevity; full tables in original notes.*

Dimensions and Dimensional Analysis

Physical Dimensions

Dimension denotes the physical nature of a quantity, such as length, mass, or time. Different units (e.g., feet, meters) can express the same dimension.

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

Dimensions and Units of Derived Quantities

Derived quantities are combinations of base quantities. Their dimensions and units are as follows:

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 to check the validity of equations by ensuring both sides have the same dimensions.

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

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

Dimension form:

• $x$ (displacement): $L$

• $a$ (acceleration): $L/T^2$

• $t$ (time): $T$

• So, $a t^2 = (L/T^2) \times T^2 = L$

Thus, both sides have dimension $L$.

Conversion of Units

Unit Conversion Factors

Conversion between SI and U.S. customary units is essential for practical applications.

• 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

Convert 38.0 m/s to mi/h:

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

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

• The driver is exceeding the speed limit of 75.0 mi/h.

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 Precision in Measurement

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 carry meaning contributing to its precision.

Rules for Significant Figures

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

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

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

• $1001$ has 4 significant figures.

Significant Figures in Calculations

Multiplying or Dividing

• The result should have the same number of significant figures as the quantity with the smallest number of 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 smallest number of decimal places.

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