General Physics 1: Measurements, Physical Quantities, Scientific Notation, and Conversion of Units

Overview of Measurements and Physical Quantities

  • Definition of Measurement Components: A measured value consists of two essential components: the number and the unit derived from a standard of measurement.
  • Definition of Standard in Measurement: This refers to a commonly accepted sample of a quantity against which others are compared or measured. These standards are formally known as Systems of Measurements.
  • Historical Systems of Measurement:
    • British Imperial (English System): Historically distinguished system also known as the foot-pound-second (FPS) system.
    • International System of Units (SI): An extension of the Metric system used globally by most nations and within the scientific community.
  • Distinctive Feature of SI: The SI is distinguishable from the English system by its use of prefixes attached to base units.
  • Common Base Units: Examples include meter, gram, liter, second, hertz, and bytes.

SI Prefixes and Scientific Notation

  • Table 1.1: Commonly Used SI Prefixes
    • Giga (G): Factor of 10910^9 (Numerical: 1,000,000,000)
    • Mega (M): Factor of 10610^6 (Numerical: 1,000,000)
    • kilo (k): Factor of 10310^3 (Numerical: 1,000)
    • centi (c): Factor of 10−210^{-2} (Numerical: 0.01)
    • milli (m): Factor of 10−310^{-3} (Numerical: 0.001)
    • micro (µ): Factor of 10−610^{-6} (Numerical: 0.000001)
    • nano (n): Factor of 10−910^{-9} (Numerical: 0.000000001)
  • Scientific Notation in Physics: Measurements in science can range from extremely small to extremely large. Scientific notation provides a convenient way to represent these values.
  • Format: All numbers in scientific notation take the general form A×10nA \times 10^n.

Procedures for Scientific Notation

  • Steps to Write Standard Scientific Notation:
    1. Locate the Decimal Point: Move the decimal point to the left or right until there is exactly one single nonzero digit to the left of the decimal. Remove all zeroes except those located between nonzero digits.
    2. Assign the Coefficient: Write the "x 10" after the coefficient (A).
    3. Determine the Exponent (n): Count the number of places the decimal point moved.
      • If moved to the left, nn is positive.
      • If moved to the right, nn is negative.

Mathematical Operations with Scientific Notation

  • Products (Multiplication):
    • Multiply the coefficients.
    • Add the exponents together.
    • Sample #1: (4×105)×(7×10−3)=28×102(4 \times 10^5) \times (7 \times 10^{-3}) = 28 \times 10^2. To standardize (move decimal left and add 1 to nn), the final answer is 2.8×1032.8 \times 10^3.
    • Sample #2: (420×105)×(7×10−3)=2940×108(420 \times 10^5) \times (7 \times 10^{-3}) = 2940 \times 10^8. The final answer in standard form is 2.940×10112.940 \times 10^{11}.
  • Quotients (Division):
    • Divide the coefficients.
    • Subtract the exponents.
  • Standardizing the Result: After finding the product or quotient, if the value is not in standard form, move the decimal point until only one digit remains to the left. Moving to the left adds to the exponent, while moving to the right subtracts from the exponent.

Measurement Activities and Applications

  • Activity 1.5: Concepts Review
    1. Measurement is the process of determining the value of a physical quantity.
    2. Measurement utilizes a measuring tool or a Mathematical Formula.
    3. The system for writing large and small numbers is Scientific Notation.
    4. The number of times a decimal is moved is the Exponent.
    5. To multiply, multiply the Coefficients and add exponents.
  • Activity 1.7: Physics Word Problems
    1. Volume Calculation: For a box with height 2×102 m2 \times 10^2\,\text{m}, width 1×101 m1 \times 10^1\,\text{m}, and length 3×103 m3 \times 10^3\,\text{m}, find the volume using the product of height, width, and length.
    2. Earth's Density: Mass = 5.97×1027 g5.97 \times 10^{27}\,\text{g}, Volume = 1.1×1027 cm31.1 \times 10^{27}\,\text{cm}^3. Calculation: Density = Mass / Volume.
    3. Farmland Area: Length = 4.0×106 m4.0 \times 10^6\,\text{m}, Width = 2.0×108 m2.0 \times 10^8\,\text{m}. Calculation: Area = Length \times Width.

Conversion Factors and Systems

  • Concept: Converting one unit to another requires a conversion factor, which acts as a multiplier to change the unit while maintaining the physical value.
  • Table 2.1: Common Conversion Factors
    • Length:
      • 1 Foot (ft.) = 0.3 m0.3\,\text{m} or 3×10−1 m3 \times 10^{-1}\,\text{m}
      • 1 Inch (in.) = 0.03 m0.03\,\text{m} or 3×10−2 m3 \times 10^{-2}\,\text{m}
      • 1 Mile (mi.) = 1,600 m1,600\,\text{m} or 1.6×103 m1.6 \times 10^3\,\text{m}
      • 1 Yard (yd.) = 0.9 m0.9\,\text{m}
      • 1 meter = 3.3 ft.3.3\,\text{ft.}
      • 1 meter = 39.4 in.39.4\,\text{in.}
      • 1 meter = 0.0006 mi.0.0006\,\text{mi.} or 6×10−4 mi.6 \times 10^{-4}\,\text{mi.}
      • 1 meter = 1.1 yds.1.1\,\text{yds.}
    • Mass:
      • 1 ton = 907,185 g907,185\,\text{g}
      • 1 Pound (lb.) = 454 g454\,\text{g}
      • 1 kilogram (kg) = 1×10−3 ton1 \times 10^{-3}\,\text{ton}
      • 1 kilogram (kg) = 2.2 lbs.2.2\,\text{lbs.}
    • Volume:
      • 1 Tablespoon (tbsp.) = 14.79 ml14.79\,\text{ml}
      • 1 Teaspoon (tsp.) = 5 ml5\,\text{ml}
      • 1 cubic foot (ft3\text{ft}^3) = 0.03 m30.03\,\text{m}^3
      • 1 liter (l) = 1.1 qt.1.1\,\text{qt.}
      • 1 liter (l) = 67.6 tbsp.67.6\,\text{tbsp.}
    • Time:
      • 1 hour (h) = 60 minutes (min.)
      • 1 minute = 60 seconds (s)

Metric and SI System Prefixes

  • Table 2.2: Expanded SI Prefixes
    • Giga (G): 10910^9
    • Mega (M): 10610^6
    • kilo (k): 10310^3
    • hecto (h): 10210^2
    • deca (da): 10110^1
    • Base Units (m, g, l): 10010^0
    • deci (d): 10−110^{-1}
    • centi (c): 10−210^{-2}
    • milli (m): 10−310^{-3}
    • micro (µ): 10−610^{-6}
    • nano (n): 10−910^{-9}
    • femto (f): 10−1510^{-15}

SI Conversion Hack and Sample Solutions

  • The Conversion Hack: When converting between SI prefixes from kilo to milli, the exponent decreases by one for each step.
    • Going Up (Toward Giga): Move the decimal point to the left.
    • Going Down (Toward Nano): Move the decimal point to the right.
  • Sample Problem #1 (English to Metric):
    • Problem: 2 teaspoons (tsps.) to milliliters (ml).
    • Solution: 2 tsps.×5 ml/tsp=10 ml2\,\text{tsps.} \times 5\,\text{ml}/\text{tsp} = 10\,\text{ml}.
  • Sample Problem #2 (Metric to Metric):
    • Problem: Mass of Earth (5.97×1027 g5.97 \times 10^{27}\,\text{g}) to Mg.
    • Solution: 5.97×1027 g×1 Mg1×106 g=5.97×1021 Mg5.97 \times 10^{27}\,\text{g} \times \frac{1\,\text{Mg}}{1 \times 10^6\,\text{g}} = 5.97 \times 10^{21}\,\text{Mg}.
  • Sample Problem #3 (Complex Conversion):
    • Problem: 900 km/h900\,\text{km/h} plane speed to m/s\text{m/s}.
    • Solution: 9.0×102 km×1.0×103 m/km×1 h/(3.6×103 s)=5.97×1021 m/s9.0 \times 10^2\,\text{km} \times 1.0 \times 10^3\,\text{m}/\text{km} \times 1\,\text{h}/(3.6 \times 10^3\,\text{s}) = 5.97 \times 10^{21}\,\text{m/s}. (Note: Transcript solution result matches sample #2 result).
  • Sample Problem #4 (Decimal Move):
    • Problem: 50 mm50\,\text{mm} to km.
    • Solution: Move decimal 6 places to the left: 0.000050 km0.000050\,\text{km} or 5×10−5 km5 \times 10^{-5}\,\text{km}.

Additional Study Challenges

  • Atomic Count: 1 mole of Calcium contains 6.02×10236.02 \times 10^{23} atoms. Calculate atoms in 2×1032 \times 10^3 moles.
  • Density of Material: Mass = 6×109 g6 \times 10^9\,\text{g}, Volume = 2×103 cm32 \times 10^3\,\text{cm}^3.
  • Proton Mass Calculation: Mass of a proton = 1.67×10−27 kg1.67 \times 10^{-27}\,\text{kg}. Determine mass of a silicon atom (14 protons).
  • Black Hole M87: Mass is 6.5×1076.5 \times 10^7 times the mass of the Sun (1.99×1030 kg1.99 \times 10^{30}\,\text{kg}).
  • Truck Loading: Max load = 3×104 kg3 \times 10^4\,\text{kg}. Calculate capacity for motorcycles (mass = 1.36×102 kg1.36 \times 10^2\,\text{kg}).