Chapter 1 (Lesson 3)

1.6 The Units of Measurement

• Units are standard quantities used to specify measurements.
• Two most common unit systems:
  • Metric system, used in most of the world (cm, m, kg, g, L, mL, km, …)
  • English system, used in the United States (in, ft, yd, lb, gal, mi, …)
• In 1960, the General Conference on Weights and Measures proposed a revised metric system for universal use by scientists.
• Seven SI Base units were formulated, with the remaining units derived from them.
• SI stands for Système International d’Unités (French origin).
• Two types of units:
  • Base units
  • Derived units

1.6.1 SI Base Units

• SI Base Units (Table 1.1):
  • Length — Symbol: m — Name: Meter
  • Mass — Symbol: kg — Name: Kilogram
  • Time — Symbol: s — Name: Second
  • Temperature — Symbol: K — Name: Kelvin
  • Amount of substance — Symbol: mol — Name: Mole
  • Electric current — Symbol: A — Name: Ampere
  • Luminous intensity — Symbol: cd — Name: Candela

The Meter: A Measure of Length

• Length is the distance between two points.
• SI unit: meter, symbol m.
• Conversions:
  • 1 ext{ m} = 1.09361 ext{ yd}
  • 1 ext{ m} = 39.37 ext{ in}
  • 1 ext{ yd} = 36 ext{ in}

The Kilogram: A Measure of Mass

• Mass is the quantity of matter within an object.
• SI unit: kilogram, symbol kg.
• Other mass units: gram (g), pound (lb).
• Conversions:
  • 1 ext{ kg} = 1000 ext{ g}
  • 1 ext{ kg} = 2.205 ext{ lb}

The Second: A Measure of Time

• Time measures the duration of an event.
• SI unit: second, symbol s.
• Time can also be expressed in minutes or hours:
  • 1 ext{ min} = 60 ext{ s}
  • 1 ext{ h} = 60 ext{ min}
  • (Note: transcript contains a typo where 1 h = 60 s is stated; correct is 60 min.)

The Kelvin: A Measure of Temperature (1 of 2)

• Temperature measures how hot or cold a substance is and relates to thermal energy.
• Kelvin (K) is the SI unit for temperature.
• The Kelvin scale starts at 0 K, the coldest possible temperature where molecular motion is assumed to stop:
  • 0 ext{ K} = -273.15^ ext{°C} = -459.67^ ext{°F}

The Fahrenheit vs Celsius vs Kelvin Relationship

• Fahrenheit degree is 5/9 the size of a Celsius degree (the size relation is 9/5 when converting C to F).
• Celsius degree and Kelvin degree are the same size.
• Temperature scale conversions:
  • ^ ext{°C} = rac{^ ext{°F} - 32}{1.8}
  • K = ^{ ext{°C}} + 273.15
  • ^ ext{°F} = rac{9}{5} ext{(^°C)} + 32
  • K = rac{^ ext{°F} + 459.67}{1.8} = (^ ext{°F} + 459.67) imes rac{5}{9}

The Kelvin: A Measure of Temperature (2 of 2)

Self-test (1) Temperature Conversion

• Problem: The conventional body temperature is approximately 310.15 ext{ K}. What is this temperature in degrees Celsius?
• Calculation:
  • ^ ext{°C} = 310.15 - 273.15 = 37.0^ ext{°C}
• Answer (correct option): 37.0^ ext{°C}

Prefix Multipliers

• Prefix multipliers (Symbol, Multiplier):
  • tera, T — 10^{12}
  • giga, G — 10^{9}
  • mega, M — 10^{6}
  • kilo, k — 10^{3}
  • deci, d — 10^{-1}
  • centi, c — 10^{-2}
  • milli, m — 10^{-3}
  • micro,  μ — 10^{-6}
  • nano, n — 10^{-9}
  • pico, p — 10^{-12}
• The SI system uses prefix multipliers with standard units to simplify large or small numbers.
• These multipliers help read, write, and communicate measurements more easily.

Conceptual Connection 1.6

• Question: Which prefix multiplier is most appropriate for reporting a measurement of 5.57 imes 10^{-5} ext{ m}?
• Answer: micro ( μ), i.e., 5.57 imes 10^{-5} ext{ m} = 55.7 ext{ μm}

1.6.2 Derived Units

• Derived units are formed from two or more of the seven base units.
• Derived quantities are quantities expressed with these units.
• Some common derived quantities and units:
  • Volume: cubic meter, ext{m}^3
  • Density: kilograms per cubic meter, rac{ ext{kg}}{ ext{m}^3}
  • Speed: meter per second, rac{ ext{m}}{ ext{s}}
  • Force: newton, ext{N} (Newton) where ext{N} = ext{kg} rac{ ext{m}}{ ext{s}^2}

Derived Units: Volume

• Volume is a measure of space and has units of length cubed:
  • ext{m}^3, ext{ cm}^3, ext{ mm}^3
• It can also be expressed in liters or gallons:
  • 1 ext{ L} = 1000 ext{ mL} = 1000 ext{ cm}^3
  • 1 ext{ mL} = 1 ext{ cm}^3
  • 1 ext{ gal}
    oughly 3.785 ext{ L}
  • 1 ext{ m}^3 = 1000 ext{ L}

Derived Units: Density (1 of 2)

• Density is the ratio of a substance’s mass to its volume:
  • 
    ho = rac{m}{V}
• Common expressions:
  ho ext{ in g/cm}^3 ext{ or g/mL}
• Conceptual significance: density determines whether a substance sinks or floats; less dense substances float on more dense substances (e.g., ice floats on water).

Derived Units: Density (2 of 2) – Example Setup

• Example setup (from transcript): A jeweler wants to know the density of a 3.75 g diamond ring.
• Procedure: Submerge ring in a graduated cylinder partly filled with water; measure water level rise.
• Formula to be used: density
  ho = rac{m}{V} where V is the displaced water volume.

Example: Diamond Ring Density (Calculation)

• Data from setup:
  • Mass, m = 3.75 ext{ g}
  • Initial water volume: 50.0 ext{ mL}
  • Final water volume: 51.07 ext{ mL}
• Displaced volume: V = 51.07 ext{ mL} - 50.0 ext{ mL} = 1.07 ext{ mL}
• Density calculation:
  • 
    ho = rac{3.75 ext{ g}}{1.07 ext{ mL}} \approx 3.50 ext{ g/mL}
• Result aligns with typical diamond density (~3.5 g/cm³).
• Note: Diamonds are denser than water, so they sink in water; the measured density confirms material identity.

Additional Notes

• Common derived units connections:
  • Volume (m³) relates to liters via 1 ext{ m}^3 = 1000 ext{ L} and 1 ext{ L} = 1000 ext{ mL}.
  • Density in SI commonly uses ext{kg/m}^3, but practical measurements often use ext{g/cm}^3 or ext{g/mL}.
• The stability of measurement relies on proper rounding and significant figures (example rounding shown at the start of the transcript).