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

Physics and Measurement

Introduction to Physics and Measurement

Physics is a science based on experimental observations and quantitative measurements. The process of measurement assigns a numerical value to a characteristic of an object or event, allowing for comparison with other objects or events.

• Physical property: Any measurable property that describes the state of a physical system. Examples include mass, length, area, volume, and velocity.

• Changes in physical properties can describe the transformations or evolution of a system between different states.

• 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. Each measurement is characterized by its type, magnitude, unit, and uncertainty.

Base quantity	Name	Symbol (SI base unit)
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 Length and Mass Scales

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

Object or Distance	Approximate Length (m)
Distance to most remote known quasar	$1.4 \times 10^{26}$
Distance from Earth to Sun	$1.5 \times 10^{11}$
Diameter of a proton	$1 \times 10^{-15}$
Diameter of a hydrogen atom	$1 \times 10^{-10}$
Diameter of a cell nucleus	$1 \times 10^{-14}$
Diameter of a human cell	$1 \times 10^{-5}$

Object	Approximate Mass (kg)
Universe	$10^{53}$
Earth	$5.98 \times 10^{24}$
Human	$10^{2}$
Electron	$9.11 \times 10^{-31}$

Dimensions and Dimensional Analysis

Dimensions

Dimension denotes the physical nature of a quantity. For example, the distance between two points can be measured in feet or meters, but both are expressions of the dimension of length.

• The symbols for the dimensions of length, mass, and time are L, M, and T, respectively.

Dimensions and Units of Derived Quantities

Quantity	Dimension	SI unit	U.S. customary unit
Area ($A$)	$L^2$	$m^2$	$ft^2$
Volume ($V$)	$L^3$	$m^3$	$ft^3$
Speed ($v$)	$L T^{-1}$	$m/s$	$ft/s$
Acceleration ($a$)	$L T^{-2}$	$m/s^2$	$ft/s^2$

Dimensional Analysis

Dimensional analysis is a method to check the correctness of equations by ensuring that both sides have the same dimensions. Only equations with consistent dimensions are physically meaningful.

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

The dimension form is:

$L = L T^{-2} T^2 = L$

This confirms the equation is dimensionally correct.

Conversion of Units

SI and U.S. Customary Units

Conversion factors are used to translate between SI and U.S. customary units:

• 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: The distance between two cities is 100 mi. What is the number of kilometers between the two cities?

• Answer: (b) larger than 100 (since 1 mi = 1.609 km, 100 mi = 160.9 km)

Example: On a highway, a car is traveling at a speed of 38.0 m/s. Is the driver exceeding the 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 with Powers of Ten

An order-of-magnitude estimate expresses a value as the nearest power of ten.

• $0.0086 m \sim 10^{-2} m$

• $0.0021 m \sim 10^{-3} m$

Significant Figures

Precision and Uncertainty

Measurements are never perfectly accurate; their values are only known within the limits of experimental uncertainty. Significant figures (or significant digits) are the digits in a number that contribute to its precision.

• 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

Rules for Significant Figures

• Multiplying or dividing: The number of significant figures in the result is the same as in 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 number of decimal places in the result should equal the smallest number of decimal places in any term in the sum or difference. Example: $135 m + 3.25 m = 138 m$ (limited by the units decimal value of 135 m)

Additional info: These foundational concepts are essential for all subsequent topics in physics, ensuring clarity and consistency in scientific communication and calculation.