Measurement and Dimensional Analysis

Measurement Accuracy and Precision

• Accuracy: The degree to which a measurement conforms to the correct or accepted value.

• Precision: The degree to which repeated measurements show the same results (consistency).

• Device Evaluation Example (for a $1.23\,\text{cm}$ rod):

  • Device A: High precision (range $0.01$), low accuracy (average $1.43$).

  • Device B: Moderate precision (range $0.03$), high accuracy (average $1.24$).

  • Device C: Low precision (range $0.07$), high accuracy (average $1.23$).

• Estimation Rule: Always record all certain digits read directly from the instrument graduations plus one estimated (uncertain) digit.

Rules for Significant Figures

• Non-zero Digits: Always significant.

• Sandwiched Zeros: Zeros between non-zero digits are significant (e.g., $502$ has $3$).

• Leading Zeros: Never significant; they are placeholders (e.g., $0.0099$ has $2$).

• Ending Zeros: Significant only if a decimal point is present (e.g., $2.00$ has $3$; $550.$ has $3$).

• Scientific Notation: Only the digits in the coefficient are counted; the $10^x$ portion is ignored.

Calculations with Significant Figures

• Addition and Subtraction: Round the result to the least precise place value (the measurement with the last significant figure furthest to the left).

• Multiplication and Division: Round the result to matches the fewest number of significant figures found in any starting measurement.

Scientific Notation

• Format: Only one non-zero digit is placed in front of the decimal.

• Positive Exponents ($10^{X}$): Represent large numbers or zeros after the number.

• Negative Exponents ($10^{-X}$): Represent small decimals or leading zeros.

• Significant Volume/Mass Constants:

  • Avogadro's Number: $6.022 \times 10^{23}\,\text{molecules/mol}$.

  • Molar Volume at STP: $22.4\,\text{L} = 1\,\text{mol}$.

Metric System and Base Units

• Properties: Decimal-based system (powers of $10$) used globally in science and engineering.

• Base Units:

  • Length: Meter ($m$).

  • Mass: Gram ($g$).

  • Volume: Liter ($L$).

  • Time: Second ($s$).

  • Temperature: Celsius ($^{\circ}\text{C}$).

  • Energy: Joule ($J$).

  • Amount: Mole ($mol$).

• Conversion: Move the decimal point based on the number of steps between prefixes (e.g., milli to hecto).

Dimensional Analysis

• Purpose: A method to convert between units using conversion factors.

• Conversion Factor: A ratio showing the relationship between two units (e.g., $1\,\text{inch} = 2.54\,\text{cm}$).

• Setup: Arrange units so that starting units are in the denominator (bottom) and target units are in the numerator (top) to allow for unit cancellation.

• Calculations:

  • $1\,\text{mile} = 1.609\,\text{km}$.

  • $1\,\text{gallon} = 3.785\,\text{L}$.