Physics uses fundamental quantities like length and mass to describe the world. These are measured in SI units like meters and kilograms. Derived quantities like speed combine fundamentals, while unit conversions and scientific notation help us work with different scales.

Measurements have limits on precision, shown by significant figures. Graphs visually represent relationships between quantities, with different shapes indicating various mathematical connections. Proper graphing techniques and interpretation skills are crucial for understanding physical phenomena.

Physical Quantities, Units, and Measurement

Physical quantities and SI units

• Fundamental quantities are basic properties of the physical world
  • Length measured in meters (m) quantifies the size or distance of an object
  • Mass measured in kilograms (kg) represents the amount of matter in an object
  • Time measured in seconds (s) quantifies the duration of an event or process
  • Electric current measured in amperes (A) represents the flow of electric charge
  • Temperature measured in kelvins (K) quantifies the average kinetic energy of particles in a substance
  • Amount of substance measured in moles (mol) represents the number of particles in a sample
  • Luminous intensity measured in candelas (cd) quantifies the brightness of a light source
• Derived quantities are combinations of fundamental quantities
  • Area ($m^2$) quantifies the size of a two-dimensional surface (square, rectangle)
  • Volume ($m^3$) measures the space occupied by a three-dimensional object (cube, sphere)
  • Density ($kg/m^3$) is the mass per unit volume of a substance (water, air)
  • Speed ($m/s$) measures the rate of change of position (car, airplane)
  • Acceleration ($m/s^2$) quantifies the rate of change of velocity (gravity, braking)
  • Force (N, $kg \cdot m/s^2$) is the push or pull on an object (friction, tension)
  • Energy (J, $kg \cdot m^2/s^2$) is the capacity to do work or cause change (kinetic, potential)
  • Power (W, $kg \cdot m^2/s^3$) measures the rate of energy transfer or conversion (light bulb, engine)
  • Pressure (Pa, $kg/(m \cdot s^2)$) is the force per unit area (atmosphere, tire)
• Unit conversions using scientific notation involve changing between different scales
  • Metric prefixes modify the base unit by powers of 10 (milli- $10^{-3}$, centi- $10^{-2}$, kilo- $10^3$, mega- $10^6$)
  • Converting between prefixes involves moving the decimal point (1 km = $10^3$ m = 1000 m)
  • Converting between different units requires multiplication or division by conversion factors (1 m/s = 3.6 km/h)
  • Dimensional analysis is a method used to check the consistency of equations and perform unit conversions

Types of Physical Quantities and Units

• Scalar quantities have only magnitude (e.g., mass, temperature, energy)
• Vector quantities have both magnitude and direction (e.g., velocity, force, displacement)
• Base units are fundamental units that cannot be derived from other units (e.g., meter, kilogram, second)
• Derived units are formed by combining base units (e.g., Newton = kg·m/s²)

Significant figures in calculations

• Significant figures convey the precision and uncertainty of a measurement
  • Non-zero digits are always significant (1, 2, 3, 4, 5, 6, 7, 8, 9)
  • Zeros between non-zero digits are significant (1.0204 has 5 significant figures)
  • Leading zeros are not significant (0.0012 has 2 significant figures)
  • Trailing zeros are significant only if the decimal point is present (1.00 has 3 significant figures, 100 has 1)
• Calculations with significant figures follow specific rounding rules
  • Addition and subtraction round the result to the least precise measurement (1.2 + 3.45 = 4.7)
  • Multiplication and division round the result to the least number of significant figures (2.5 × 3.42 = 8.6)
• Measurement uncertainty relates to the precision of the measuring instrument
  • Significant figures reflect the uncertainty in a measurement (ruler marked in mm vs. cm)
• Accuracy refers to how close a measurement is to the true value
• Precision refers to the reproducibility of measurements and is reflected in the number of significant figures

Graphs of physical relationships

• Types of graphs represent different mathematical relationships between variables
  • Linear graphs show a constant rate of change (position vs. time for constant velocity)
    • Slope represents the rate of change (velocity)
    • y-intercept represents the initial value (initial position)
  • Inverse graphs have a hyperbolic shape (volume vs. pressure for an ideal gas at constant temperature)
    • Product of x and y coordinates is constant (Boyle's law: $PV = k$)
  • Quadratic graphs have a parabolic shape (position vs. time for constant acceleration)
    • Vertex represents the turning point (maximum height)
  • Logarithmic graphs use a logarithmic scale on one or both axes (sound intensity level vs. sound intensity)
    • Represents exponential relationships as linear (decibel scale)
• Creating graphs involves proper formatting and labeling
  • Choose appropriate axes and scales to fit the data range
  • Label axes with physical quantities and units (Time (s), Position (m))
  • Plot data points accurately using a consistent scale
  • Draw a best-fit line or curve to represent the trend
• Interpreting graphs requires analyzing the relationship between variables
  • Determine the relationship between variables based on the shape of the graph (linear, inverse, quadratic)
  • Calculate slopes and intercepts to quantify rates of change and initial values
  • Extrapolate or interpolate values beyond or between measured data points