College Physics - Chapter 1 Lecture Notes

Chapter 1 Lecture Notes: Models, Measurements, and Vectors

Goals

  • Understand standards and units; perform unit conversions.
  • Express measurements and calculations with the correct number of significant figures.
  • Add and subtract vectors graphically and analytically.
  • Resolve vectors into x- and y-components.

Measurement

  • Physics relies on experimental science.
    • Observe natural phenomena.
    • Make predictions using:
      • Models
      • Hypotheses
      • Theories
      • Laws

Units of Measurement

  • Cultural units (e.g., cubit, span, foot, mile) vary with time and location.
  • In 1889, the General Conference on Weights and Measures established the Système International.
    • Refer to Appendix A for details.

Three Fundamental S.I. Units

  • Time: Second (s)
  • Length: Meter (m)
  • Mass: Kilogram (kg)

The Second

  • Originally defined based on the length of a day.
  • Now defined with high accuracy using an atomic clock.
    • Based on 9,192,631,770 oscillations of a low-energy transition in Cesium (Cs).
    • Occurs in the microwave region.

The Meter

  • Original definition (1791): 1/10,000,000 of the distance from the North Pole to the Equator.
  • More recent definition: tied to Krypton discharge and counting wavelengths.
  • Current definition: the distance light travels in a vacuum in 1/299,792,458 seconds.
    • Loses only 1 second in 30 million years.

The Kilogram

  • Defined by a reference cylinder kept in Sevres, France.
  • Seeking a more modern, atomic reference.

Adjusting Fundamental Units: Powers of 10

  • Use prefixes to adjust unit sizes by powers of ten (see Table 1.1).
    • Example: Measuring the distance from San Francisco to Charlotte in kilometers instead of meters.
    • Distance: 4,621 kilometers or 4.6 Millimeter.

Table 1.1: Prefixes for Powers of 10

  • 10−1810^{-18}: atto- (a)
  • 10−1510^{-15}: femto- (f)
  • 10−1210^{-12}: pico- (p)
  • 10−910^{-9}: nano- (n)
  • 10−610^{-6}: micro- (\mu)
  • 10−310^{-3}: milli- (m)
  • 10−210^{-2}: centi- (c)
  • 10310^{3}: kilo- (k)
  • 10610^{6}: mega- (M)
  • 10910^{9}: giga- (G)
  • 101210^{12}: tera- (T)
  • 101510^{15}: peta- (P)
  • 101810^{18}: exa- (E)

Conversions

  • Practice is essential.
  • Convert to meters, kilograms, and seconds for consistency in calculations.
  • Address two main challenges:
    • Derived units
    • English to S.I. conversions

Derived Units

  • Example: Energy is measured in Joules (J).
    • J=kg∗m2s2J = kg * \frac{m^2}{s^2}
    • Errors in mass or distance affect the calculation.

English → S.I.

  • Common English units: miles, feet, pounds, quarts, gallons.
  • Memorize key conversions:
    • Displacement
    • Volume
    • Mass
  • Mass is tricky because English units relate kilograms to pounds using standard Earth gravity, even though kilograms measure mass and pounds measure force.

English → S.I. Two

  • Helpful conversions:
    • Displacement: 2.54 centimeters = 1 inch
    • Mass: 454 grams = 1 pound
    • Volume: 1 liter = 1.06 quarts

Unit Conversion Example

  • Alpha Centauri is 4.3 light-years away.
  • Convert this distance to kilometers.
  • Use the relationship: Distance = time × speed
  • The speed is the speed of light.

Precision and Significant Figures

  • Measurement tools have limitations.
  • Report results reflecting these limitations.

Am I Significant?

  • Avoid reporting excessive digits from calculators.
    • Example: 10 (1 SF) / 3 (1 SF) = 3.33333333 (from calculator).
    • Reporting the calculator's result implies high precision.
    • Inaccurate reporting can cause practical problems, e.g., bolt holes not aligning.

Vector Addition

  • In vector addition, 1 + 1 does not always equal 2.
  • Can be performed graphically or by components.

Vector Components

  • Decompose vectors into components for solving.
  • A=A<em>x+A</em>yA = A<em>x + A</em>y
  • Ax=Acos⁡(θ)A_x = A \cos(\theta)
  • Ay=Asin⁡(θ)A_y = A \sin(\theta)
  • A=A<em>x2+A</em>y2A = \sqrt{A<em>x^2 + A</em>y^2}
  • tan(θ)=A<em>yA</em>xtan(\theta) = \frac{A<em>y}{A</em>x}

Trigonometry Review

  • Review trigonometry concepts.
  • Try Example 1.6 to check understanding.

Using Components to Add Vectors

  • Example 1.7: Vector has magnitude 50 cm, direction 30°; vector has magnitude 35 cm, direction 110°.
  • Find the resultant vector.