Kinematics: Units and Unit Conversion

Converting Units and Measurement Systems

Standard Measurements in Kinematics

  • Three Common Measurements: In kinematics, the fundamental quantities are:

    • Length

    • Time

    • Mass

  • Derived Quantities: All other quantities in kinematics are derived from these three basic measurements.

  • Importance of Units: To properly understand a measurement, its unit must be known. For example, 120 lbs120 \, \text{lbs} is vastly different from 120 kgs120 \, \text{kgs}.

    • NASA Mars Climate Orbiter Example: On September 23, 1999, NASA lost a 125125 million spacecraft due to a unit conversion error. Lockheed Martin used English units, while NASA used SI units, leading to the failure.

Systems of Measurement


  • There are three primary systems of measurement:

    • SI (Système International): Also known as the metric or MKS system (Meter-Kilogram-Second).

    • CGS: Centimeter-Gram-Second system.

    • English: Foot-Pound-Second system (though a slug is used for mass).


  • Basic Units for Kinematics:

    Quantity

    SI (Metric/MKS)

    CGS

    English


    Length

    Meter (m)

    Centimeter (cm)

    Foot (ft)


    Time

    Second (s)

    Second (s)

    Second (s)


    Mass

    Kilogram (kg)

    Gram (g)

    Slug

    • Universal Constants: These units are defined in terms of universal constants, such as the speed of light, ensuring they can be consistently measured across all countries.

    The SI System and Prefixes

    • Standard SI Prefixes: These prefixes indicate powers of 1010, making it easy to express very large or very small quantities.

      • Rarely Used Prefixes: Yotta (Y), Zetta (Z), Exa (E), hecto (h), deca (da), atto (a), zepto (z), and yocto (y) are mentioned as rarely used.

    • Examples of SI Prefix Conversions:

      • 1 nanometer=10−9 m1 \, \text{nanometer} = 10^{-9} \, \text{m}

      • 1 micrometer=10−6 m1 \, \text{micrometer} = 10^{-6} \, \text{m}

      • 1 millimeter=10−3 m1 \, \text{millimeter} = 10^{-3} \, \text{m}

      • 1 kilometer=103 m1 \, \text{kilometer} = 10^{3} \, \text{m}

      • 1 megavolt=106 V1 \, \text{megavolt} = 10^{6} \, \text{V}

      • 1 gigahertz=109 Hz1 \, \text{gigahertz} = 10^{9} \, \text{Hz}

    Converting Units

    • Metric-to-Metric Conversions: These are straightforward as they only involve powers of 1010. For example, converting from kilograms to grams.

    • Metric-to-British Conversions: These typically require more effort due to less direct relationships.

    • Conversion Method using Equalities:

      • Use equality ratios written as a value of "1" to change units.

      • For example, since 1 mi=1610 m1 \, \text{mi} = 1610 \, \text{m}, we can write conversion factors as rac1 mi1610 m=1rac{1 \, \text{mi}}{1610 \, \text{m}} = 1 or rac1610 m1 mi=1rac{1610 \, \text{m}}{1 \, \text{mi}} = 1.

      • Crucial Step: Always ensure that units cancel out correctly to arrive at the desired final unit.

    • Conversion Examples:

      • Meters to Feet: Given 1 m=3.28084 ft1 \, \text{m} = 3.28084 \, \text{ft}, an 8611-m8611 \text{-m} mountain is converted to feet as:
        8611 m×3.28084 ft1 m=28251 ft8611 \, \text{m} \times \frac{3.28084 \, \text{ft}}{1 \, \text{m}} = 28251 \, \text{ft}

      • Minutes to Seconds: Convert minutes to seconds using the equality 1 min=60 s1 \, \text{min} = 60 \, \text{s}:
        X min×60 s1 min=(X×60) sX \, \text{min} \times \frac{60 \, \text{s}}{1 \, \text{min}} = (X \times 60) \, \text{s}

      • Centimeters to Inches: Given 1 inch=2.54 cm1 \, \text{inch} = 2.54 \, \text{cm}, to convert 5 cm5 \, \text{cm} to inches:
        5 cm×1 inch2.54 cm≈1.9685 inches5 \, \text{cm} \times \frac{1 \, \text{inch}}{2.54 \, \text{cm}} \approx 1.9685 \, \text{inches}

      • Inches to Centimeters: Convert 5 inches5 \, \text{inches} to centimeters:
        5 inches×2.54 cm1 inch=12.7 cm5 \, \text{inches} \times \frac{2.54 \, \text{cm}}{1 \, \text{inch}} = 12.7 \, \text{cm}

      • SI Conversion Example (cm to mm): How many mm are in 25 cm25 \, \text{cm}?

        • We know 1 cm=10−2 m1 \, \text{cm} = 10^{-2} \, \text{m} and 1 mm=10−3 m1 \, \text{mm} = 10^{-3} \, \text{m}.

        • 25 cm×10−2 m1 cm×1 mm10−3 m=25×10−2×103 mm=25×101 mm=250 mm25 \, \text{cm} \times \frac{10^{-2} \, \text{m}}{1 \, \text{cm}} \times \frac{1 \, \text{mm}}{10^{-3} \, \text{m}} = 25 \times 10^{-2} \times 10^{3} \, \text{mm} = 25 \times 10^{1} \, \text{mm} = 250 \, \text{mm}.

        • Alternatively, using the Slide's calculation: 25 cm=25×10−2 m25 \, \text{cm} = 25 \times 10^{-2} \, \text{m}. To convert to mm, we need to divide by 10−3 m/mm10^{-3} \, \text{m/mm}. So, rac25×10−2 m10−3 m/mm=25×10−2−(−3) mm=25×101 mm=2.5×102 mmrac{25 \times 10^{-2} \, \text{m}}{10^{-3} \, \text{m/mm}} = 25 \times 10^{-2-(-3)} \, \text{mm} = 25 \times 10^{1} \, \text{mm} = 2.5 \times 10^2 \, \text{mm} .

      • Centimeters to Kilometers: Convert 5 cm5 \, \text{cm} to km.

        • 5 cm×1 m100 cm×1 km1000 m=5×10−5 km=0.00005 km5 \, \text{cm} \times \frac{1 \, \text{m}}{100 \, \text{cm}} \times \frac{1 \, \text{km}}{1000 \, \text{m}} = 5 \times 10^{-5} \, \text{km} = 0.00005 \, \text{km}

    Converting Areas and Volumes

    • Squared/Cubed Conversion Factors: If a unit is squared (for area) or cubed (for volume), the conversion factor must also be squared or cubed, respectively.

    • Example: Liter to Cubic Centimeters:

      • 1 liter=1 L=10 cm×10 cm×10 cm=1000 cm31 \, \text{liter} = 1 \, \text{L} = 10 \, \text{cm} \times 10 \, \text{cm} \times 10 \, \text{cm} = 1000 \, \text{cm}^3

    • Problem Example: Cubic Inches in a 2.0L Bottle: How many cubic inches are in a 2.0 L2.0 \, \text{L} bottle of soda, given 1 in=2.54 cm1 \, \text{in} = 2.54 \, \text{cm}?

      • First, convert liters to cubic centimeters: 2.0 L=2.0×1000 cm3=2000 cm32.0 \, \text{L} = 2.0 \times 1000 \, \text{cm}^3 = 2000 \, \text{cm}^3

      • Next, convert cubic centimeters to cubic inches. Since 1 in=2.54 cm1 \, \text{in} = 2.54 \, \text{cm}, then (1 in)3=(2.54 cm)3(1 \, \text{in})^3 = (2.54 \, \text{cm})^3, so 1 in3=(2.54)3 cm3≈16.387 cm31 \, \text{in}^3 = (2.54)^3 \, \text{cm}^3 \approx 16.387 \, \text{cm}^3

      • Finally, 2000 cm3×1 in316.387 cm3≈122.04 in32000 \, \text{cm}^3 \times \frac{1 \, \text{in}^3}{16.387 \, \text{cm}^3} \approx 122.04 \, \text{in}^3

    Multiple Step Conversion Example

    • Problem: How many hours in 5125 \frac{1}{2} weeks?

      • 5.5 weeks×7 days1 week×24 hours1 day=5.5×7×24 hours=924 hours5.5 \, \text{weeks} \times \frac{7 \, \text{days}}{1 \, \text{week}} \times \frac{24 \, \text{hours}}{1 \, \text{day}} = 5.5 \times 7 \times 24 \, \text{hours} = 924 \, \text{hours}

    PRETEST Group Volume Conversion Problem

    • Conversion Ratios Provided:

      • 2 jiggers=1 jack2 \, \text{jiggers} = 1 \, \text{jack}

      • 2 jacks=1 gill2 \, \text{jacks} = 1 \, \text{gill}

      • 2 gills=1 cup2 \, \text{gills} = 1 \, \text{cup}

      • 2 cups=1 pint2 \, \text{cups} = 1 \, \text{pint}

      • 2 pints=1 quart2 \, \text{pints} = 1 \, \text{quart}

      • 2 quarts=1 pottle2 \, \text{quarts} = 1 \, \text{pottle}

      • 2 pottles=1 gallon2 \, \text{pottles} = 1 \, \text{gallon}

      • 2 gallons=1 pail2 \, \text{gallons} = 1 \, \text{pail}

      • 2 pails=1 peck2 \, \text{pails} = 1 \, \text{peck}

      • 4 pecks=1 bushel4 \, \text{pecks} = 1 \, \text{bushel}

      • 2 bushels=1 strike2 \, \text{bushels} = 1 \, \text{strike}

      • 2 strikes=1 coomb2 \, \text{strikes} = 1 \, \text{coomb}

      • 2 coombs=1 cask2 \, \text{coombs} = 1 \, \text{cask}

      • 2 casks=1 barrel2 \, \text{casks} = 1 \, \text{barrel}

      • 2 barrels=1 hogshead2 \, \text{barrels} = 1 \, \text{hogshead}

      • 2 hogsheads=1 pipe2 \, \text{hogsheads} = 1 \, \text{pipe}

      • 2 pipes=1 tun2 \, \text{pipes} = 1 \, \text{tun}

    • Question: How many POTTLES are there in 5 STRIKES5 \, \text{STRIKES}?

      • We need to set up a conversion chain from strikes to pottles:

        • 1 strike=2 bushels1 \, \text{strike} = 2 \, \text{bushels}

        • 1 bushel=4 pecks1 \, \text{bushel} = 4 \, \text{pecks}

        • 1 peck=2 pails1 \, \text{peck} = 2 \, \text{pails}

        • 1 pail=2 gallons1 \, \text{pail} = 2 \, \text{gallons}

        • 1 gallon=2 pottles1 \, \text{gallon} = 2 \, \text{pottles}

      • Combine these conversion factors:
        1 strike×2 bushels1 strike×4 pecks1 bushel×2 pails1 peck×2 gallons1 pail×2 pottles1 gallon1 \, \text{strike} \times \frac{2 \, \text{bushels}}{1 \, \text{strike}} \times \frac{4 \, \text{pecks}}{1 \, \text{bushel}} \times \frac{2 \, \text{pails}}{1 \, \text{peck}} \times \frac{2 \, \text{gallons}}{1 \, \text{pail}} \times \frac{2 \, \text{pottles}}{1 \, \text{gallon}}

      • 1 strike=(2×4×2×2×2) pottles=64 pottles1 \, \text{strike} = (2 \times 4 \times 2 \times 2 \times 2) \, \text{pottles} = 64 \, \text{pottles}

      • For 5 strikes5 \, \text{strikes}, the number of pottles is:
        5 strikes×64pottlesstrike=320 pottles5 \, \text{strikes} \times 64 \frac{\text{pottles}}{\text{strike}} = 320 \, \text{pottles}

    Friday's Class Instructions

    • Preparation: Watch Lecture 2a before class.

    • In-Class Activity: There will be a Socrative concept problem quiz. Students should download the Socrative app and sign in for the class (Stampe2053).

    • Attendance: Students who were absent the previous Monday must ensure their attendance is recorded for iRattler.