Key Formulas

Percentage Uncertainty

$$\text{percentage uncertainty} = \frac{\text{absolute uncertainty}}{\text{measured value}} \times 100\%$$

Converts the margin of doubt into a proportion of the measurement. A larger measured value gives a smaller percentage uncertainty for the same instrument.

Percentage Difference

$$\text{percentage difference} = \frac{|\text{experimental value} - \text{accepted value}|}{\text{accepted value}} \times 100\%$$

Compares your experimental result to a known or accepted value. If the percentage difference is less than the total percentage uncertainty, the result is consistent with the accepted value.

Addition / Subtraction -- Add Absolute Uncertainties

$$\text{If } R = A + B \text{ or } R = A - B, \quad \Delta R = \Delta A + \Delta B$$

When quantities are added or subtracted, their absolute uncertainties are added together.

Multiplication / Division -- Add Percentage Uncertainties

$$\text{If } R = A \times B \text{ or } R = \frac{A}{B}, \quad \%\Delta R = \%\Delta A + \%\Delta B$$

When quantities are multiplied or divided, their percentage uncertainties are added together.

Power Rule -- Multiply Percentage Uncertainty by the Power

$$\text{If } y = x^n, \quad \%\Delta y = n \times \%\Delta x$$

When a quantity is raised to a power, the percentage uncertainty is multiplied by the power. For example, if $y = x^2$, the percentage uncertainty in $y$ is double that of $x$.

Gradient of a Straight Line

$$\text{gradient} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$$

Use a large triangle on the line of best fit. The gradient often has physical significance (e.g. spring constant from an $F$ vs $x$ graph).

Straight-Line Equation

$$y = mx + c$$

Where $m$ is the gradient and $c$ is the y-intercept. Rearrange experimental equations into this form to plot straight-line graphs and extract physical constants.

Frequency from Time Period

$$f = \frac{1}{T}$$

Used when reading time periods from oscilloscope displays. Measure the time period $T$ from the waveform, then calculate frequency.

Mean Value

$$\bar{x} = \frac{\sum x_i}{n}$$

Sum all readings and divide by the number of data points. When calculating a mean, it is acceptable to increase significant figures by one compared to the raw data.