Uncertainties and Methods

SI Units and Their Prefixes

• Fundamental SI Units include:

  • Mass (m): kilogram (kg)

  • Length (l): metre (m)

  • Time (t): second (s)

  • Electric Current (I): ampere (A)

  • Temperature (T): kelvin (K)

  • Amount of Substance (n): mole (mol) Units for derived quantities, like force and voltage, can be obtained from these base units. For example:

  • Force $(F = ma)$: kg·m/s² or N (newton)

  • Voltage $(V = E/Q)$ : derived as $V = \frac{kg \cdot m^2}{A \cdot s^3}$

• SI Prefixes scale units by powers of ten. Common prefixes include:
  • kilo (k): $10^3$
  • mega (M): $10^6$
  • milli (m): $10^{-3}$
  • micro (µ): $10^{-6}$

Limitation of Physical Measurements

Types of Errors:

1. Random Errors:

• Affect precision and cause scattered readings.
• Example: electronic noise in measurement.
• Minimising Random Errors:
• Take multiple measurements and calculate the mean.
• Use digital equipment (e.g., data loggers).
• Choose appropriate tools (e.g., a micrometer for small lengths).

2. Systematic Errors:

• Affect accuracy by consistently skewing results in one direction.
• Example: zero error on a balance.
• Reducing Systematic Errors:
• Calibrate instruments regularly.
• Correct for background radiation in radiation measurements.
• Measure liquids at eye level to avoid parallax errors.

Types of Uncertainty:

• Absolute Uncertainty: Fixed uncertainty $($e.g., $\pm 0.5 \, \text{mm} )$.
• Fractional Uncertainty: Absolute uncertainty divided by the measurement $($e.g., $\frac{0.5}{10} )$.
• Percentage Uncertainty: Fractional uncertainty as a percentage $($e.g., $\frac{0.5}{10} \times 100\% )$.

Calculating and Reducing Uncertainty

• Reduce Percentage/Fractional Uncertainty by measuring larger quantities when possible (e.g., a longer rope).

• Uncertainty in a Reading: $\pm$ half the smallest scale division (e.g.,$\pm 0.5 \, \text{°C}$) for a thermometer with $1°C$ divisions).

• Uncertainty for Multiple Readings: $\pm$ half the range (e.g., if measurements range from $62$ to $64$, the uncertainty is $\pm 1$). Combining Uncertainties:

• Addition/Subtraction: Add absolute uncertainties.

• Multiplication/Division: Add percentage uncertainties.

• Raising to a Power: Multiply percentage uncertainty by the power.

Uncertainties and Graphs

• Represent uncertainties with error bars on graphs.

• A line of best fit should go through all error bars (excluding outliers). Finding Uncertainty in Gradients:

• Draw steepest and shallowest lines within error bars.

• Calculate the difference in gradients to find percentage uncertainty:

$$\text{Percentage uncertainty} = \frac{\text{best gradient - worst gradient}}{\text{best gradient}} \times 100\%$$

---

Estimation of Physical Quantities

• Orders of Magnitude: Power of ten approximations for size comparison (e.g., nucleus diameter is around $10^{-14} \, \text{m}$). Experiment Design:

• Variables:

  • Independent: What you change.
  • Dependent: What you measure.
  • Control: Variables that stay constant.

• Data Types:

  • Discrete: Specific values (e.g., count of objects).
  • Continuous: Range of values (e.g., temperature).
  • Categorical: Qualitative data (e.g., colour).

Graphing Relationships:

• Linear Relationships: For $y = kx$, plot $y$ against $x$ for a straight line.
• Reciprocal Relationships: For $y = \frac{k}{x}$, plot $y$ against $\frac{1}{x}$.

Logarithmic and Exponential Functions:

• Convert equations like $R = kA^n$ into a linear form by taking logarithms:

$$\ln R = n \ln A + \ln k$$

Plotting $\ln R$ against $\ln A$ gives a straight line with gradient $n$.