Measurements and Problem Solving

Chapter 2 Description:

  • On December 11, 1998, NASA launched the Mars Climate Orbiter, the first weather satellite for a planet other than Earth.

  • The mission failed due to a mix-up with measurement units; commands sent from Earth used English units (pound-seconds) instead of the metric standard (Newton-seconds), causing the Orbiter to enter the Martian atmosphere at an altitude that was too low, ultimately resulting in the Orbiter's disintegration.

  • The failure cost NASA around $125 million.

  • Quote by Sir William Lawrence Bragg:
    "The important thing in science is not so much to obtain new facts as to discover new ways of thinking about them."

Learning Outcomes

  • 2.1 Express very large and very small numbers using scientific notation.

  • 2.2 Report measured quantities to the right number of digits.

  • 2.3 Determine which digits in a number are significant.

  • 2.4 Round numbers to the correct number of significant figures.

  • 2.5 Determine the correct number of significant figures in the results of multiplication and division calculations.

  • 2.6 Determine the correct number of significant figures in the results of addition and subtraction calculations.

  • 2.7 Determine the correct number of significant figures in the results of calculations involving both addition/subtraction and multiplication/division.

  • 2.8 Recognize and work with the SI base units of measurement, prefix multipliers, and derived units.

  • 2.9 Convert between units.

  • 2.10 Convert units in a quantity that has units in the numerator and the denominator.

The Metric Mix-Up

Findings:

  • The navigational error of the Mars Climate Orbiter was due to commands sent in English units without conversion to metric standards.

  • This led to an incorrect trajectory and the Orbiter entering the Martian atmosphere at about 35 miles (57 kilometers) instead of the intended 87 to 93 miles (140 to 150 kilometers).

  • The measurements for adjustments were 4.45 times too small, failing to maintain a sufficiently high altitude.

Definition:

  • A unit is a standard, agreed-upon quantity by which other quantities are measured.

Scientific Notation

2.2 Key Concepts and Operations

  • Positive exponent: Indicates multiplication by 10 raised to an integer power (ex: 10^3 = 1000).

  • Negative exponent: Indicates division by 10 raised to an integer power (ex: 10^{-3} = 0.001).

  • Structure of Scientific Notation:

    • Comprised of two parts: a decimal part (between 1 and 10) and an exponential part (10 raised to an exponent).

Conversion Examples in Scientific Notation

  • $1.0 x 10^5 = 100,000$

  • $6.7 x 10^3 = 6700$

  • $1.0 x 10^{-6} = 0.000001$

  • $6.7 x 10^{-3} = 0.0067$

Process to Convert to Scientific Notation

  1. Move the decimal point to form a number between 1 and 10.

  2. Multiply that number by 10 raised to a power that corresponds to how many places the decimal is moved.

    • Positive exponent results from leftward movement.

    • Negative exponent results from rightward movement.

Example - U.S. Population in Scientific Notation

  • The 2016 U.S. population: $323,000,000 = 3.23 x 10^8$ people.

Conceptual Checkpoint 2.1

  • The radius of a dust speck is $6.5 x 10^{-3}$ mm.

    • Decimal notation options presented:

    • a. 6500 mm

    • b. 0.065 mm

    • c. 0.0065 mm

    • d. 0.00065 mm

Significant Figures & Precision

2.3 Concepts

  • The last reported digit in a measurement signifies the uncertainty in that measurement.

  • Example: Global temperature average rise of 0.7 °C reported as 0.6 °C signifies 0.7 ± 0.1 °C, indicating a potential range of 0.6 °C to 0.8 °C. The precision affects decisions, such as those in policy-making.

Reporting Digits and Precision

  • Significant Figure Rules:

    1. All nonzero digits are significant.

    2. Interior zeros between significant digits are significant.

    3. Trailing zeros in a decimal number are significant.

    4. Leading zeros are not significant.

    5. Trailing zeros before an implied decimal point are ambiguous.

Example of Rounding to Significant Figures

  • When rounding:

    • 2.33 to two significant figures becomes 2.3

    • 2.37 becomes 2.4

    • 2.349 becomes 2.3.

Additional Checkpoint

  • Example 2.2:
    The density of a platinum ring is tested to determine if it is genuine with comparisons against known values (density of platinum = 21.4 g/cm³).

Units of Measurement

2.5 Overview

  • There are three systems of units: English, metric, and SI (International System of Units).

  • Base Units include:

    • Length: Meter

    • Mass: Kilogram

    • Time: Second

Specific SI Definitions

  • Length: Meter defined as the distance light travels in a vacuum during rac{1}{299792458} seconds.

  • Mass: Kilogram defined as a specific mass of a block of metal.

  • Time: Second, defined as the duration of 9,192,631,770 periods of radiation emitted from the specified transition in a cesium-133 atom.

Conversion Factors with Prefixes

  • Used for practical measuring.

  • Relation examples:

    • 1 kg = 10^3 g

    • 1 cm = 10^{-2} m

    • 1 ns = 10^{-9} s

Understanding Density

  • Density (d) is calculated as: d = rac{m}{V} , where:

    • m = mass (g)

    • V = volume (cm³ or mL)

Problem Solving Strategies

2.6 Approach

  • Unit Conversion: Many problems involve converting quantities from one unit to another using dimensional analysis.

  • Solution Map: Visual outline showing the necessary steps for problem-solving; diagramming unit relationships aids in conversion.

  • Procedure: Sort information, strategize steps, solve mathematically, and check for physical sense and correctness of units.

Example of Unit Conversion

  • Convert 0.825 m to millimeters:

    • Relationships used: 1 mm = 10^{-3} m.

Multistep Unit Conversion Example

  • Converting gas mileage from miles/gallon to kilometers/liter involves multiple factors, ensuring clarity and correctness through each transition.

Density as a Numerical Property

2.10 Importance

  • Density is essential for characterizing and identifying materials.

  • Example: For a sample with a mass of 27.2 g and a volume of 22.5 mL:

    • Density Calculation:
      d = rac{27.2 ext{ g}}{22.5 ext{ mL}} = 1.21 ext{ g/mL}

Testing Density of Materials

  • Any sample's density can be compared against known densities to ascertain material composition.

Conclusion and Summary

Chapter 2 Review

  • Emphasizes significant figures, precision in measurements, proper unit conversions, and understanding density as crucial parts of scientific understanding and problem-solving.

  • Units employed promote clarity in communication and enhance measurement accuracy in experiments and observations, applying these concepts across various scientific disciplines.