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^{-18}: atto- (a)
• 10^{-15}: femto- (f)
• 10^{-12}: pico- (p)
• 10^{-9}: nano- (n)
• 10^{-6}: micro- (\mu)
• 10^{-3}: milli- (m)
• 10^{-2}: centi- (c)
• 10^{3}: kilo- (k)
• 10^{6}: mega- (M)
• 10^{9}: giga- (G)
• 10^{12}: tera- (T)
• 10^{15}: peta- (P)
• 10^{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 * \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 = Ax + Ay
• A_x = A \cos(\theta)
• A_y = A \sin(\theta)
• A = \sqrt{Ax^2 + Ay^2}
• tan(\theta) = \frac{Ay}{Ax}

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.