Comprehensive Notes on Measurement and Physical Science Fundamentals of Physical Science

Measurement and the Physical Sciences

• Science is a description that leads to an understanding of our environment. This description fundamentally involves the measurement of the physical world.

• Understanding the environment demands the interpretation of accurate measurements, which are referred to as data. Consequently, a comprehensive understanding of measurement is essential to scientific study.

• Sophisticated methods of measurement have been developed to track movement, temperature, weather conditions, time, and other physical variables.

• A critical limitation exists in measurement: as smaller and smaller objects are measured, it becomes apparent that the act of measuring actually distorts the object. This implies that not everything can be measured with absolute certainty.

• Physical Science is a subset of the Natural Sciences, distinct from Biological Sciences. It encompasses five primary disciplines:

  • Physics

  • Chemistry

  • Geology

  • Meteorology

  • Astronomy

Scientific Investigation and Methodology

• Measurements serve as the foundation for all scientific research and investigation. Phenomena are observed in nature, which leads to questions regarding how or why they occur.

• Scientists operate under the assumption that the universe is orderly and can be understood.

• The Scientific Method consists of general methods of observations, rules for reasoning, and making predictions. It is broken down into specific stages:

  • Observations and Measurements: The stage where quantitative data are gathered.

  • Hypothesis: A possible explanation for the observations; it is a tentative answer or an educated guess. An example is the idea that matter consists of small particles (atoms) that simply rearrange themselves. New experiments are designed specifically to test the validity of a hypothesis. A hypothesis is supported if it correctly predicts experimental results.

  • Experiments: The process of testing under controlled conditions to determine if results support or confirm the hypothesis. Experimental results must be capable of being duplicated by other researchers. No concept or model of nature is considered valid unless its predictions agree with experimental results.

  • Theory: A tested explanation for a broad segment of basic natural phenomena. For instance, Atomic Theory has withstood testing for over $200$ years. Theories may be accepted, modified, or rejected based on continued experimentation.

  • Scientific Law: A concise statement in words or mathematical form that describes a fundamental relationship of nature after a series of experiments. An example is the Law of Conservation of Mass, which states there is no gain or loss of mass during a chemical reaction. A law states the finding but does not explain the behavior.

The Role of the Senses and Instrumentation

• Humans utilize five senses to perceive the environment: sight, hearing, smell, taste, and touch.

• Sight and hearing provide the brain with the most information regarding the environment.

• Every sense has limitations that can be reduced through the use of measuring devices. Instruments extend the human ability to measure and learn more about the environment.

• Senses are not always reliable and can be deceived, as demonstrated by various optical illusions.

Standard Units and Systems of Measurement

• Measurements are expressed in two parts: magnitude and units.

• Fundamental quantities required for the study of force and motion are length, mass, and time.

• A Standard Unit is a fixed and reproducible value used to take accurate measurements.

• There are two major systems of units:

  • British (English) System: Widely used primarily in the United States. Units include miles, inches, pounds, and seconds.

  • Metric System: Used throughout most of the world. Units include kilometers, meters, and grams. The U.S. officially adopted the metric system in $1893$, though it continues to rely on the British system.

• Length is the measurement of space in any direction. Space consists of three dimensions: length, width, and height.

  • Metric Standard Unit: Meter ($m$). Originally defined as $\frac{1}{10,000,000}$ of the distance from the equator to the North Pole.

  • British Standard Unit: Foot. Originally referenced to the size of the human foot.

• Mass is the amount of matter an object contains and is a fundamental unit that remains constant throughout the universe.

  • Metric Standard Unit: Kilogram ($kg$). Originally defined as the amount of water in a $0.1\,m$ cube. It is currently referenced to a standard cylinder kept in Paris.

• Weight and British Mass:

  • The British Standard Unit for mass is the Slug, though it is rarely used.

  • The Pound ($lb$) is commonly used in the British system, but it is a unit of weight, not mass. Weight is related to gravitational attraction and depends on the object's location. For example, an object weighing $1\,lb$ on Earth would weigh approximately $\frac{1}{6}\,lb$ on the Moon. Weight varies slightly on Earth based on altitude (higher altitude results in less weight).

• Time is the continuous, forward-flowing of events and has only one direction (forward).

  • Standard Unit: Second ($s$). This is the standard in both the metric and British systems.

  • Historical Definition: $\frac{1}{86,400}$ of a solar day.

  • Modern Definition: Based on the vibration of the $Cs^{133}$ atom, measured by an Atomic Clock.

The Metric System and SI Base Units

• The Metric System is often called the "mks system" after its standard units: meter, kilogram, and second.

• It is a decimal (base-10) system, making it more convenient than the British system. It is administered by the Bureau International des Poids et Mesures (BIPM) in Paris.

• The International System of Units (SI) contains seven base units, which are regarded as dimensionally independent:

  • Meter ($m$) for length

  • Kilogram ($kg$) for mass

  • Second ($s$) for time

  • Ampere ($A$) for electrical current

  • Kelvin ($K$) for temperature

  • Mole ($mol$) for the amount of a substance

  • Candela ($cd$) for luminous intensity

Metric Prefixes and Volume Relationships

• Base-10 conversions are easy and convenient compared to the British system:

  • Metric: $1\text{ kilometer} = 1000\text{ meters}$; $1\text{ meter} = 100\text{ centimeters}$; $1\text{ liter} = 1000\text{ milliliters}$.

  • British: $1\text{ mile} = 5280\text{ feet}$; $1\text{ yard} = 3\text{ feet}$ or $36\text{ inches}$; $1\text{ quart} = 32\text{ ounces}$; $1\text{ gallon} = 128\text{ ounces}$.

• Commonly used prefixes include:

  • Mega ($M$): $10^{6}$ ($1,000,000$ times the base)

  • Kilo ($k$): $10^{3}$ ($1,000$ times the base)

  • Centi ($c$): $10^{-2}$ ($\frac{1}{100}\text{th}$ of the base)

  • Milli ($m$): $10^{-3}$ ($\frac{1}{1000}\text{th}$ of the base)

• Volume and the Liter (Nonstandard Metric Unit):

  • A Liter ($L$) is defined as the volume of liquid in a $0.1\,m$ ($10\,cm$) cube. Calculation: $10\,cm \times 10\,cm \times 10\,cm = 1000\,cm^3$.

  • A liter of pure water has a mass of $1\,kg$ ($1000\,g$).

  • Therefore, $1\,cm^3$ (or $cc$) of pure water has a mass of $1\,g$.

  • By definition: $1\,L = 1000\,mL$; thus, $1\,mL = 1\,cc = 1\,g$ of pure water.

  • While $1\,mL = 1\,cc$ for all liquids, other liquids will not necessarily have a mass of $1\,g$.

  • Comparison: A Liter is slightly more than a quart. $1\text{ quart} = 0.946\text{ liters}$; $1\text{ liter} = 1.06\text{ quarts}$.

• Metric Ton: The mass of $1\text{ cubic meter}$ ($1\,m^3$) of water.

  • Since $1\,m = 100\,cm$, then $(100\,cm)^3 = 1,000,000\,cm^3$.

  • Given $1000\,cm^3 = 1\,L$, there are $1000\,L$ in $1\,m^3$.

  • With each liter having a mass of $1\,kg$, the metric ton equals $1\,kg \times 1000 = 1000\,kg$.

Derived Units and Density

• It is difficult to measure everything using only the seven fundamental units; therefore, derived units (multiples or combinations of fundamental units) are used.

• Examples of derived units:

  • Volume: $\text{length}^3$ ($m^3$, $cm^3$)

  • Area: $\text{length}^2$ ($m^2$, $ft^2$)

  • Speed: $\frac{\text{length}}{\text{time}}$ ($m/s$, $mi/h$)

• Density ($\rho$) is the mass per unit volume: $\rho = \frac{m}{V}$. It measures how compact a substance is.

  • Units: $g/cm^3$ or $kg/m^3$.

  • Specific densities: Aluminum ($Al$) = $2.7\,g/cm^3$; Iron ($Fe$) = $7.8\,g/cm^3$; Gold ($Au$) = $19.3\,g/cm^3$.

  • The average density for the solid Earth is $5.5\,g/cm^3$.

• Liquid Densities can be measured using a Hydrometer (a weighted glass bulb). The higher the hydrometer floats, the greater the density of the liquid.

  • Pure water = $1\,g/cm^3$.

  • Seawater = $1.025\,g/cm^3$.

  • Urine = $1.015$ to $1.030\,g/cm^3$.

  • Hydrometers are used to test antifreeze in car radiators by measuring the density of the liquid.

• Complex Unit Combinations:

  • Newton ($N$) = $kg \times m/s^2$

  • Joule ($J$) = $kg \times m^2/s^2$

  • Watt ($W$) = $kg \times m^2/s^3$

Unit Conversion Procedures

• Conversion factors relate one unit to another. They are used to convert between the British and Metric systems or within a single system.

• Example factors: $1\text{ inch} = 2.54\text{ centimeters}$; $1\,mi/h = 1.61\,km/h$; $1\,km/h = 0.621\,mi/h$; $1\,yd = 0.914\,m$.

• Steps to convert:

  • Step 1: Choose an appropriate conversion factor.

  • Step 2: Arrange the factor into a form where unwanted units cancel out.

• Multi-step conversions (e.g., inches to meters) involve converting through intermediate units (e.g., inches to centimeters to meters).

Significant Figures and Rounding Rules

• Significant Figures (SF) are a method for expressing measured numbers properly.

• Rule of precision: A mathematical operation cannot result in more significant figures than the starting numbers.

• General Rule for calculation: Report only as many SF in the result as there are in the quantity with the least significant figures (the limiting term).

  • Example: $\frac{6.8\,cm}{1.67\,cm} = 4.1$ (rounded from $4.0718563$ because $6.8$ has only two SF).

  • Example: $5.687 + 11.11 = 16.80$ (rounded from $16.797$ because $11.11$ has four SF).

• Deterministic Rules for SF:

  • All non-zero digits are significant ($23.4$ and $234$ both have $3$ SF).

  • Zeros between non-zero digits ("captive zeros") are significant ($20.05$ has $4$ SF; $407$ has $3$ SF).

  • Zeros to the left of non-zero digits ("leading zeros") are not significant; they only locate decimal points ($0.0000035$ has $2$ SF).

  • Trailing zeros (to the right of all non-zero digits) must be determined from context ($45.0$ has $3$ SF, but $4500$ likely has only $2$ SF).

  • Exact numbers (counts of people or items) are assumed to have an unlimited number of SF.

• Rounding Rules:

  • If the first digit to be dropped is less than $5$, leave the preceding digit unchanged (e.g., $26.142$ rounds to $26.1$ for $3$ SF).

  • If the first digit to be dropped is $5$ or greater, increase the preceding digit by one (e.g., $10.063$ rounds to $10.1$ for $3$ SF).

  • Special rounding: $0.0997$ rounds to $0.10$ for two SF.

Scientific Notation (Powers-of-10)

• Scientific notation is used for very large or very small numbers. For example, $1,000,000 = 10^{6}$.

• Distance to the sun is preferentially formatted as $9.3 \times 10^{7}\text{ miles}$.

• Shifting the decimal point:

  • Shifting left increases the exponent by one for every place shifted ($360,000 = 3.6 \times 10^{5}$).

  • Shifting right decreases the exponent by one for every place shifted ($0.0694 = 6.94 \times 10^{-2}$).

• Logic in calculation:

  • $5.6256 \times 0.0012 = 0.0067507$. Rounding to two SF yields $0.0068$, which is $6.8 \times 10^{-3}$.

  • $\frac{0.0024}{8.05} = 0.0002981$. Rounding to two SF yields $0.00030$, which is $3.0 \times 10^{-4}$. Note that the zero is significant for preserving two SF.

Scientific Problem Solving and Numerical Applications

• A standardized approach to solving problems involves six steps:

  1. Read the problem and identify the relevant chapter principle.

  2. Write down given quantities with their units.

  3. Make a sketch of the scenario.

  4. Determine what is wanted and write it down.

  5. Check units and perform necessary conversions.

  6. Survey equations, select the appropriate one, perform the math, round off, and adjust to the correct number of significant figures.

• Application Example (Earth's Orbit):

  • The Earth orbits the Sun in a nearly circular path with a radius ($r$) of $93\text{ million miles}$.

  • Goal: Find the distance traveled in one revolution (circumference, $c$).

  • Formula: $c = 2\pi r$, where $\pi \approx 3.14159$ .

  • Calculation: $c = 2 \times 3.14159 \times 93,000,000\text{ miles}$, or $c = 2 \times 3.14159 \times 9.3 \times 10^{7}\text{ miles}$.

  • Raw Result: $58.433574 \times 10^{7}\text{ miles}$.

  • Adjusted for SF: $c = 5.8 \times 10^{8}\text{ miles}$ (rounded to two significant figures to match the input of $93$ million).