Measurement and Uncertainties

1.1 Measurements in Physics

Fundamental and Derived Units

• Fundamental SI Units:
  • Mass: Kilogram (kg)
  • Distance: Meter (m)
  • Time: Second (s)
  • Electric current: Ampere (A)
  • Amount of substance: Mole (mol)
  • Temperature: Kelvin (K)
• Derived Units: Combinations of fundamental units like:
  • Velocity: m/s
  • Force: Newton (N = kg*m/s²)
  • Energy: Joule (J = kg*m²/s²)

Scientific Notation and Metric Multipliers

• Scientific Notation: Values in form a×10^n (1 ≤ a < 10, n is integer)
  • Speed of light: 3×10^8 m/s
  • Centimeter: 1×10^-2 m
• Metric Multipliers:
  • Peta (P): 10^15
  • Tera (T): 10^12
  • Giga (G): 10^9
  • Mega (M): 10^6
  • Kilo (k): 10^3
  • Milli (m): 10^-3
  • Micro: 10^-6
  • Nano (n): 10^-9

Significant Figures

• Rules of Significance:
  • All figures are significant except leading zeros and trailing zeros without a decimal.
  • Multiplication/Division: Result should match least precise value's significant figures.
  • Addition/Subtraction: Result should match least decimal place.

Orders of Magnitude

• Orders of magnitude use powers of 10 to compare sizes:
  • Diameter of the observable universe: 10^26 m
  • Mass of the universe: 10^53 kg
  • Time: Age of universe: 10^17 s

Estimation

• Estimations are to the nearest power of 10, as shown in orders of magnitude examples.

1.2 Uncertainties and Errors

Random and Systematic Errors

• Random Errors:
  • Caused by measurement fluctuations around true value.
  • Reduced by averaging.
• Systematic Errors:
  • Fixed shifts in measurements, not reducible by averaging.
  • Caused by biases.

Absolute, Fractional, and Percentage Uncertainties

• Definitions:
  • Absolute Uncertainty: ±x
  • Fractional Uncertainty: x/x
  • Percentage Uncertainty: (x/x)×100%
• Calculations with Uncertainties:
  • Addition/Subtraction: Sum of absolute uncertainties.
  • Multiplication/Division: Sum of fractional uncertainties.
  • Powers: Fractional uncertainty × |n|.

Error Bars

• Indicate uncertainties on graphs, vertical/horizontal, total length of two absolute uncertainties.

Uncertainty of Gradient and Intercepts

• Line of Best Fit: Minimizes average distance between data points and line.
• Maximum/Minimum Line: Max/min slopes passing all error bars.
• Uncertainty in Intercept/Gradient: Difference between line of best fit and max/min lines.

1.3 Vectors and Scalars

Vector and Scalar Quantities

• Scalar: Defined by magnitude only (e.g., distance, speed).
• Vector: Defined by magnitude and direction (e.g., displacement, velocity).

Combination and Resolution of Vectors

• Vector Addition/Subtraction:
  • Methods: Parallelogram and head to tail.
  • Closed polygon vectors add up to zero.
• Vector Resolution:
  • Vectors can be resolved into perpendicular components.

FYI

• Graphical relationship determination:
  • Linear relationships and transformations,
  • Slope and intercept analysis.