Measurement Systems

Overview of Measurement Systems

Measurement systems provide standardized units for quantifying physical properties. The three main systems are the English System, Metric System, and SI System (International System of Units).

• English System: Used mainly in the United States; units include inches, feet, pounds, and gallons. Not based on multiples of 10, making conversions less straightforward.

• Metric System: Used in most countries; units include meters, grams, and liters. Based on powers of 10 for easy conversions.

• SI System: A standardized version of the metric system used globally by scientists. Base units include meter (length), kilogram (mass), and second (time).

Units of Measurement

Base Units and Derived Units

Physical quantities are measured using base units, and derived units are formed from these base units.

• Length: Distance between two points (SI base unit: meter).

• Mass: Amount of matter in an object (SI base unit: kilogram).

• Time: Duration of an event (SI base unit: second).

• Temperature: Average kinetic energy of particles (SI base unit: kelvin).

• Volume: Space occupied by a substance (1 mL = 1 cm3).

Metric Unit Prefixes

Metric prefixes indicate multiples or fractions of base units.

Temperature

Temperature Scales and Conversions

Temperature measures the average kinetic energy of particles. Common scales include Fahrenheit (°F), Celsius (°C), and Kelvin (K). Kelvin is an absolute scale with no negative values.

• Kelvin (K):

• Celsius (°C):

• Fahrenheit (°F):

Example: If the weather is 29°C, then and K.

Density

Definition and Properties

Density is the mass of an object divided by its volume. It is an intensive property (independent of the amount of substance) and a physical property (can be used to identify substances).

• Density formula:

• Units: g/L for gases, g/mL for liquids, g/cm3 for solids

• Density typically decreases with increasing temperature.

• Order: solids > liquids >>> gases (with exceptions, e.g., ice is less dense than liquid water).

Densities of Common Materials (at 20°C)

Volume Displacement Method

Used to measure the density of irregular solids:

1. Find the mass using an analytical balance.

2. Find the volume by water displacement:

Measurements

Exact vs. Measured Numbers

• Exact numbers: Known with complete certainty (e.g., counting objects, defined values such as 1 inch = 2.54 cm).

• Measured numbers: Obtained using measuring devices; always include some uncertainty.

Every measurement includes:

• A number (size or magnitude)

• A unit (standard of comparison)

• An indication of uncertainty (last digit is estimated)

Significant Figures

Rules for Counting Significant Figures

• Always significant: Nonzero digits, trailing zeros (after decimal), captive zeros (between significant digits)

• Never significant: Leading zeros, placeholder zeros (at the end of a whole number without a decimal)

Examples:

• 456 (3 sig figs)

• 12.340 (5 sig figs)

• 0.0045 (2 sig figs)

• 100.0 (4 sig figs)

Scientific Notation

Expresses numbers as a coefficient (1 ≤ x < 10) times a power of 10. All digits in the coefficient are significant.

• For numbers > 1, power of 10 is positive (e.g., )

• For numbers < 1, power of 10 is negative (e.g., )

Rounding and Calculations with Significant Figures

• Rounding: If the next digit is 5 or greater, round up; if less than 5, leave as is.

• Addition/Subtraction: Round to the least number of decimal places.

• Multiplication/Division: Round to the least number of significant figures.

Accuracy and Precision

Definitions

• Accuracy: How close a measurement is to the accepted value.

• Precision: How consistent repeated measurements are.

All measurements should be within ±1 of the last certain digit to be considered accurate or precise.

Dimensional Analysis

Problem Solving with Units

Dimensional analysis (factor-label method) uses units to guide problem solving. Conversion factors are ratios of equivalent quantities expressed in different units.

1. Write the known quantity and unit.

2. Multiply by conversion factors so units cancel.

3. Continue until you reach the desired unit.

Common Conversion Factors

Example: Density Calculation Using Dimensional Analysis

Given: 4.00 qt sample of antifreeze weighs 9.26 lb. Find density in g/mL.

• Convert pounds to grams:

• Convert quarts to liters:

• Convert liters to milliliters:

• Density:

Knowledge Check and Review Questions

• Practice identifying units, significant figures, and performing conversions.

• Apply concepts to real-world and laboratory scenarios (e.g., density of objects, temperature conversions, accuracy and precision analysis).

Additional info: These notes are foundational for scientific measurement and problem-solving, which are essential for all science and engineering courses, including Precalculus applications involving units, conversions, and quantitative reasoning.