Measuring stuff is key in physics. We use a system of units to describe the world around us. The SI system gives us seven fundamental units, like meters for length and seconds for time, that form the basis for all measurements.

From these basic units, we can create more complex ones. For example, we combine mass, length, and time to get newtons for force. Understanding how to convert between units and measure accurately is crucial for solving physics problems and understanding the world.

Fundamental Units and Measurement Systems

Fundamental and derived SI units

• Fundamental SI units form basis of measurement system
  • Meter (m) measures length of path traveled by light in vacuum in 1/299,792,458 second
  • Kilogram (kg) defines mass using Planck constant (6.62607015 × 10^-34 J⋅s)
  • Second (s) duration of 9,192,631,770 periods of radiation from cesium-133 atom
  • Ampere (A) flow of 1/(1.602176634 × 10^-19) elementary charges per second
  • Kelvin (K) sets 1 K as 1/273.16 of thermodynamic temperature of triple point of water
  • Mole (mol) contains exactly 6.02214076 × 10^23 elementary entities (Avogadro constant)
  • Candela (cd) measures luminous intensity in a given direction
• Derived SI units combine fundamental units
  • Newton (N) force required to accelerate 1 kg mass at 1 m/s^2 ($N = kg \cdot m/s^2$)
  • Joule (J) work done when 1 N force displaces object 1 m ($J = N \cdot m$)
  • Watt (W) rate of energy transfer of 1 J per second ($W = J/s$)
  • Pascal (Pa) pressure exerted by 1 N force over 1 m^2 area ($Pa = N/m^2$)
  • Volt (V) electric potential difference when 1 J of work moves 1 C of charge ($V = J/C$)
  • Ohm (Ω) electrical resistance between two points when 1 V produces 1 A current ($Ω = V/A$)
• Prefixes for SI units simplify large or small measurements
  • Kilo (k) multiplies by 1000 (kilometer)
  • Mega (M) multiplies by 1,000,000 (megawatt)
  • Giga (G) multiplies by 1,000,000,000 (gigabyte)
  • Milli (m) divides by 1000 (millimeter)
  • Micro (μ) divides by 1,000,000 (microgram)
  • Nano (n) divides by 1,000,000,000 (nanosecond)

Unit conversion techniques

• Length conversions involve ratios and multiplication
  • 1 meter equals 100 centimeters facilitates cm to m conversion
  • 1 kilometer contains 1000 meters useful for long distances
  • 1 inch precisely defined as 2.54 centimeters aids US-metric conversions
  • 1 mile approximately 1.609 kilometers important for travel distances
• Mass conversions relate different weight scales
  • 1 kilogram subdivides into 1000 grams for smaller masses
  • 1 pound converts to 0.4536 kilograms bridges US-metric systems
• Time conversions based on sexagesimal system
  • 1 minute comprises 60 seconds fundamental time unit
  • 1 hour contains 3600 seconds used in rate calculations
  • 1 day consists of 86400 seconds relevant for daily phenomena
• Temperature conversions between scales
  • Celsius to Fahrenheit multiplies by 9/5, adds 32 ($°F = (°C × \frac{9}{5}) + 32$)
  • Celsius to Kelvin adds 273.15 ($K = °C + 273.15$) absolute zero reference
  • Fahrenheit to Celsius subtracts 32, multiplies by 5/9 ($°C = (°F - 32) × \frac{5}{9}$)

Precision and accuracy in measurements

• Precision reflects consistency of repeated measurements
  • Determined by smallest scale division on measuring instrument (ruler markings)
  • High precision measurements cluster closely together (dart throws near each other)
• Accuracy indicates closeness of measured value to true value
  • Affected by systematic errors (miscalibrated scale)
  • High accuracy measurements center around true value (dart throws near bullseye)
• Significant figures convey measurement certainty
  • All non-zero digits carry meaning (123 has 3 significant figures)
  • Zeros between non-zero digits significant (1002 has 4 significant figures)
  • Leading zeros not significant (0.00123 has 3 significant figures)
  • Trailing zeros after decimal point significant (1.2300 has 5 significant figures)
• Calculations with significant figures preserve measurement certainty
  • Addition and subtraction: result has same decimal places as least precise measurement
  • Multiplication and division: result has same number of significant figures as least precise measurement

Dimensional analysis applications

• Dimensions of physical quantities express fundamental units
  • Length [L] measures spatial extent (meter)
  • Mass [M] quantifies amount of matter (kilogram)
  • Time [T] represents duration (second)
  • Derived quantities combine dimensions (velocity [LT^-1], force [MLT^-2])
• Dimensional homogeneity ensures equation validity
  • Both sides of equation must have same dimensions
  • Used to check for errors in complex formulas
• Steps for dimensional analysis:
  1. Identify dimensions of all quantities in equation
  2. Express derived quantities in terms of fundamental dimensions
  3. Verify dimensional consistency across equation
• Deriving relationships using dimensional analysis:
  1. Identify relevant physical quantities in problem
  2. Combine quantities to form dimensionless groups
  3. Use these groups to construct possible relationships between variables
• Applications extend beyond equation checking
  • Estimating scaling laws in physical systems (fluid dynamics)
  • Simplifying complex problems by reducing variables (wind tunnel testing)