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

Physics and Measurement

Introduction to Physics and Measurement

Physics is a natural science that seeks to understand the fundamental laws governing the universe. It relies on experimental observations and quantitative measurements to describe and predict physical 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 are measurable attributes whose values describe the state of a physical system.

• Examples of physical properties include mass, length, area, volume, and velocity.

• Changes in physical properties can be used to describe transformations or evolutions of a system.

• Physical properties are often referred to as observables.

SI Units

International System of Units (SI)

Measurements in physics most commonly use the International System of Units (SI) as a standardized framework for comparison. SI units provide a consistent basis for scientific communication and calculation.

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 span a wide range of values. Below are examples of typical lengths and masses encountered in physics.

Measured Length	Approximate Value (m)
Distance from Earth to most remote known quasar	$1.4 \times 10^{26}$
Distance from Earth to nearest large galaxy (Andromeda)	$2 \times 10^{22}$
Diameter of a hydrogen atom	$1 \times 10^{-10}$
Diameter of a proton	$1 \times 10^{-15}$
Diameter of a person	$1 \times 10^{0}$

Object	Approximate Mass (kg)
Universe	$1 \times 10^{53}$
Earth	$5.98 \times 10^{24}$
Human	$7 \times 10^{0}$
Electron	$9.11 \times 10^{-31}$

Dimensional Analysis

Dimensions and Units

Dimension denotes the physical nature of a quantity. Different units (such as feet, meters) can express the same dimension (e.g., length). The symbols used to specify the dimensions of length, mass, and time are L, M, and T, respectively.

Quantity	Dimension	SI Unit	U.S. Customary Unit
Area	$L^2$	$m^2$	$ft^2$
Volume	$L^3$	$m^3$	$ft^3$
Speed	$L/T$	$m/s$	$ft/s$
Acceleration	$L/T^2$	$m/s^2$	$ft/s^2$

Dimensional Analysis in Equations

Dimensional analysis is used to check the validity of equations by ensuring that both sides have the same dimensions. This is a fundamental requirement for physical relationships.

• Any equation is only correct if the dimensions on both sides match.

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

Conversion of Units

Unit Conversion Factors

Conversion between SI and U.S. customary units is often necessary in physics. The following are common conversion factors for length:

• 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 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 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.

• Significant figures indicate the reliability of a measured value.

• Examples:

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

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

  • $1001$ has 4 significant figures.

Rules for Significant Figures in Calculations

Multiplying or Dividing

• The number of significant figures in the final answer is the same as the number in the quantity with the smallest number of significant figures.

• Example: $25.57 m \times 2.45 m = 62.6 m^2$ (result 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: $135 m + 3.25 m = 138 m$ (result limited to the units decimal value by 135 m).

Quiz Example

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

  • (a) smaller than 100

  • (b) larger than 100

  • (c) equal to 100

  Answer: (b) larger than 100

Additional info: This study guide covers foundational concepts in physics measurement, SI units, dimensional analysis, and significant figures, which are essential for all subsequent topics in college physics.