Scientific Measurement

Why We Measure

• Measurement: The process of assigning a numerical value to a physical quantity through quantitative comparison to a standard unit. It allows for objective description and analysis.

• Essential for science, engineering, health, and daily life because it enables reproducibility of experiments, precise design, accurate diagnostics, and common understanding of quantities.

• Three kinds of numbers in measurement:

  • Counted (Exact): Numbers obtained by counting discrete items; they have no uncertainty and possess an infinite number of significant figures (e.g., 12 apples, 5 testing samples).

  • Defined (Exact, may be fractional): Numbers that come from definitions or conversion factors; they also have infinite significant figures (e.g., $1000 \, \text{g} = 1 \, \text{kg}$, $1 \, \text{inch} = 2.54 \, \text{cm}$ exactly).

  • Measured (from instruments): Numbers obtained from using measuring devices; these always have some degree of uncertainty and their precision is limited by the instrument itself.

Accuracy vs. Precision

• Accuracy: Refers to how close a measured value is to the true or accepted value. A measurement is accurate if it is very close to what it should be. Imagine hitting the bullseye on a dartboard.

• Precision: Refers to the consistency or reproducibility of a series of repeated measurements. Precise measurements cluster closely together, even if they are far from the true value. Imagine consistently hitting the same spot on a dartboard, even if it's not the bullseye.

• Combinations: It's crucial to evaluate both accuracy and precision:

  • Good accuracy, good precision: Measurements are both close to the true value and close to each other.

    • Visual Example: On a dartboard, all darts are tightly clustered around the bullseye.

    • Detailed Explanation: This is the ideal scenario where the measuring system is both correctly calibrated and consistently repeatable. The data points are tightly grouped in the center of the target.

  • Poor accuracy, good precision: Measurements are consistent but systematically incorrect (e.g., due to a faulty instrument). They are close to each other but far from the true value.

    • Visual Example: On a dartboard, all darts are tightly clustered, but far away from the bullseye (e.g., all stuck in the top-left corner).

    • Detailed Explanation: This indicates a systematic error. The instrument or method consistently gives the wrong result, but it does so reproducibly. Calibration or method adjustment is needed.

  • Good accuracy, poor precision: Measurements are spread out but their average might be close to the true value. The individual measurements are inconsistent.

    • Visual Example: On a dartboard, darts are scattered widely across the board, but some are close to the bullseye, and their average position might be near the center.

    • Detailed Explanation: This suggests random errors or variability in the measurement process. Despite the wide spread, the measurements might 'average out' to the correct value, but any single measurement is unreliable. Efforts should focus on reducing variability.

  • Poor accuracy, poor precision: Measurements are inconsistent and far from the true value.

    • Visual Example: On a dartboard, darts are scattered widely and randomly all over the board, with no clear cluster near the bullseye.

    • Detailed Explanation: This is the least desirable outcome, showing both systematic and random errors. The measuring system is neither calibrated correctly nor repeatable, leading to unreliable and meaningless results.

Reading Instruments

• When reading any analog measuring instrument, record all digits that are known for certain (from the markings) plus one additional digit that is estimated.

• The estimated digit accounts for the uncertainty in reading between the smallest marked divisions.

• Examples for a ruler with different levels of precision:

  • Ruler marked in $0.1 \, \text{m}$ increments: A reading of $0.6 \, \text{m}$ means the $0.6$ is certain, but the next digit is implied to be zero or simply not precise enough to estimate.

  • Ruler marked in $0.01 \, \text{m}$ increments: A reading of $0.61 \, \text{m}$ means the $0.6$ and $0.01$ are certain, and the final digit (e.g., $0.007$).

  • Ruler marked in $0.001 \, \text{m}$ increments: A reading of $0.607 \, \text{m}$ means the $0.60$ are certain, and the $0.007$ is estimated.

Significant Figures (Sig Figs)

Significant figures indicate the precision of a measurement, including all certain digits and one estimated digit. Rules for determining significant figures:

1. All non-zero digits are significant.

   • Example: $28.35 \, \text{g}$ has 4 sig figs.

2. Leading zeros are never significant. These zeros only act as placeholders for the decimal point.

   • Example: $0.0028 \, \text{L}$ has 2 sig figs (the leading zeros before the '2' are not significant).

3. Captive zeros (zeros between non-zero digits) are always significant.

   • Example: $1005 \, \text{kg}$ has 4 sig figs.

4. Trailing zeros (at the end of the number):

   • Significant if the number contains a decimal point. These indicate precision.

     • Example: $15.00 \, \text{cm}$ has 4 sig figs.

     • Example: $250.0 \, \text{mL}$ has 4 sig figs.

   • Not significant if the number does not contain an explicitly written decimal point (implied decimal point). These are often just placeholders.

     • Example: $1200 \, \text{m}$ has 2 sig figs.

     • To make trailing zeros significant, a decimal point must be added. Example: $1200.$ has 4 sig figs.

5. Exact counts & definitions have an unlimited number of significant figures.

   • Example: 1 dozen eggs (exactly 12 eggs, unlimited sig figs), $1 \, \text{m} = 100 \, \text{cm}$ (by definition, unlimited sig figs).

Rounding Rules

• Addition/Subtraction: The result must be rounded to the same number of decimal places as the measurement with the fewest decimal places in the calculation.

  • Principle: The answer cannot be more precise than the least precise measurement in terms of its decimal places.

  • Example: Calculate $12.52 \, \text{cm} + 349.0 \, \text{cm} + 8.243 \, \text{cm}$

    • $12.52 \, \text{cm}$ (2 decimal places)

    • $349.0 \, \text{cm}$ (1 decimal place)

    • $8.243 \, \text{cm}$ (3 decimal places)

    • The least precise measurement is $349.0 \, \text{cm}$ with 1 decimal place.

    • Sum: $12.52 + 349.0 + 8.243 = 369.763 \, \text{cm}$

    • Rounded to 1 decimal place: $369.8 \, \text{cm}$

• Multiplication/Division: The result must be rounded to the same number of significant figures as the measurement with the fewest significant figures in the calculation.

  • Principle: The answer's precision is limited by the measurement with the fewest significant figures.

  • Example: Calculate $6.221 \, \text{cm} \times 5.2 \, \text{cm}$

    • $6.221 \, \text{cm}$ (4 sig figs)

    • $5.2 \, \text{cm}$ (2 sig figs)

    • The least precise measurement is $5.2 \, \text{cm}$ with 2 significant figures.

    • Product: $6.221 \times 5.2 = 32.3492 \, \text{cm}^2$

    • Rounded to 2 significant figures: $32 \, \text{cm}^2$

Scientific Notation

• Format: Expresses very large or very small numbers compactly: $a \times 10^{n}$, where $a$ (the coefficient) is a number greater than or equal to 1 and less than 10 (1 \le a < 10), and $n$ (the exponent) is an integer.

• Purpose: Used to represent numbers with appropriate significant figures and to simplify calculations with extremely large or small quantities.

• Exponent ($n$) Interpretation:

  • Positive $n$: Indicates a large number (decimal point moved $n$ places to the right).

  • Example: $602,200,000,000,000,000,000,000 = 6.022 \times 10^{23}$

  • Negative $n$: Indicates a small number (decimal point moved $n$ places to the left).

  • Example: $0.0000000000000000000000166 = 1.66 \times 10^{-24}$

• Operations with Scientific Notation:

  • Add/Subtract: Exponents ($n$) must be equalized first before adding or subtracting the coefficients ($a$).

  • Example: $(2.5 \times 10^3) + (1.2 \times 10^4) = (0.25 \times 10^4) + (1.2 \times 10^4) = 1.45 \times 10^4$

  • Multiply: Multiply the coefficients ($a$ values) and add the exponents ($n$ values).

  • Example: $(3.0 \times 10^2) \times (2.0 \times 10^4) = (3.0 \times 2.0) \times 10^{(2+4)} = 6.0 \times 10^6$

  • Divide: Divide the coefficients ($a$ values) and subtract the exponents ($n$ values).

  • Example: $(6.0 \times 10^5) \div (2.0 \times 10^2) = (6.0 \div 2.0) \times 10^{(5-2)} = 3.0 \times 10^3$

SI Units & Prefixes

• The International System of Units (SI) is the modern form of the metric system and is the world's most widely used system of measurement.

• Base units: Fundamental units from which all other units are derived. They are considered independent.

  • m: meter (length)

  • kg: kilogram (mass)

  • s: second (time)

  • K: Kelvin (thermodynamic temperature)

  • A: Ampere (electric current)

  • mol: mole (amount of substance)

  • cd: candela (luminous intensity)

• Key Prefixes: Multipliers applied to base units to denote larger or smaller quantities by powers of ten.

  • k = kilo = $10^{3}$ (e.g., 1 kilometer = $10^3$ meters)

  • c = centi = $10^{-2}$ (e.g., 1 centimeter = $10^{-2}$ meters)

  • m = milli = $10^{-3}$ (e.g., 1 milliliter = $10^{-3}$ liters)

  • $\mu$ = micro = $10^{-6}$ (e.g., 1 microsecond = $10^{-6}$ seconds)

  • n = nano = $10^{-9}$ (e.g., 1 nanometer = $10^{-9}$ meters)

  • p = pico = $10^{-12}$ (e.g., 1 picogram = $10^{-12}$ grams)

Derived Quantities

• Derived quantities are physical quantities that are expressed as algebraic combinations of base quantities.

• They have units that are derived from the SI base units through multiplication or division.

  • Area $= \text{length}^2$ (units like $\text{m}^2$, $\text{cm}^2$)

  • Volume $= \text{length}^3$ (units like $\text{m}^3$, $\text{cm}^3$)

  • Speed $= \dfrac{\text{distance}}{\text{time}}$ (units like $\text{m/s}$)

  • Density $= \dfrac{\text{mass}}{\text{volume}}$ (units like $\text{kg/m}^3$, $\text{g/cm}^3$)

Unit Conversions

• The process of changing a measurement from one unit to another while maintaining its value.

• Dimensional Analysis: A systematic approach using conversion factors to change units. Conversion factors are ratios equal to one that convert between units (e.g., $\frac{1 \, \text{in}}{2.54 \, \text{cm}}$ or $\frac{2.54 \, \text{cm}}{1 \, \text{in}}$).

• Metric ⇆ Metric: Conversions involve moving the decimal point according to the difference in powers of ten indicated by the prefixes.

• Metric ⇆ English: Require specific fixed conversion factors (e.g., $1 \, \text{in} = 2.54 \, \text{cm}$ exactly, $1 \, \text{lb} = 453.6 \, \text{g}$).

• Maintain Sig Figs: Ensure the final answer reflects the precision of the original measurements; exact conversion factors do not limit significant figures.

Density Problems

• Density ($\rho$): An intrinsic physical property of a substance, defined as mass per unit volume. It is unique for each substance at a given temperature and pressure.

• Formulas:

  • $\rho = \dfrac{m}{V}$ (density = mass / volume)

  • $m = \rho V$ (mass = density \times volume)

  • $V = \dfrac{m}{\rho}$ (volume = mass / density)

• Cylinder volume: For objects with regular geometric shapes, specific volume formulas are used. For a cylinder: $V = \pi r^{2}h$ (where $r$ is the radius and $h$ is the height).

Temperature Scales & Conversions

• Temperature is a measure of the average kinetic energy of the particles in a substance. Higher kinetic energy means higher temperature.

• Common Scales: Each scale has different reference points for freezing and boiling water.

  • Celsius ($\text{\degree C}$ ): Widely used globally for everyday temperatures and in scientific contexts. Water freezes at $\text{0{}C}$ and boils at $\text{100C}$ at standard atmospheric pressure. The Celsius scale is a centigrade scale, meaning there are 100 degrees between the freezing and boiling points of water.

  • Fahrenheit ($\text{\degree F}$ ): Primarily used in the United States for everyday temperatures. Water freezes at $\text{32{}F}$ and boils at $\text{212{}F}$ . There are 180 degrees between these two points.

  • Kelvin (K): The absolute temperature scale used in scientific applications. Zero Kelvin ($0 \, \text{K}$ or $-273.15\,\text{{}C}$ ) is absolute zero, where all molecular motion theoretically stops. There are no negative temperatures on the Kelvin scale.

  • Kelvin degrees are the same size as Celsius degrees.

• Conversion Formulas: Allows for conversion between the three scales.

  • $\text{K}=\text{ {}C}+273.15$

  • $\text{F}=1.8\times\text{ C}+32$

  • $\text{{}C}=\dfrac{\text{{}F}-32}{1.8}$