diff --git a/docs/sph3u7.md b/docs/sph3u7.md index 1b08cc0..3a8dc28 100644 --- a/docs/sph3u7.md +++ b/docs/sph3u7.md @@ -6,7 +6,7 @@ The course code for this page is **SPH3U7**. ### Fundamental units -Every other SI unit is derived from fundamental units. Memorise these! +Every other SI unit is derived from the fundamental SI units. Memorise these! | Quantity type | Unit | Symbol | | --- | --- | --- | @@ -59,7 +59,7 @@ Every SI unit can be expanded with metric prefixes. $$1.2 × 2.0 = 2.4$$ !!! warning - When rounding an answer with significant figures, if the **least significant figure** is $5$, round up only if the **second-least** significant figure is odd. + When rounding an answer with significant figures, if the **least significant figure** is $5$, round up only if the **second-least** significant figure is **odd**. $$1.25 + 1.2 = 2.4$$ $$1.35 + 1.2 = 2.6$$ @@ -76,11 +76,77 @@ Scientific notation is written in the form of $m×10^{n}$, where $1 \leq m < 10, The order of magnitude of a number can be found by converting it to scientific notation and taking its power of 10. !!! example - - The order of magnitude of 212000, or $2.12×10^{5}$, is 5. - - The order of magnitude of 0.212, or $2.12×10^{-1}$, is -1. + - The order of magnitude of $212000$, or $2.12×10^{5}$, is 5. + - The order of magnitude of $0.212$, or $2.12×10^{-1}$, is -1. ## 1.2 - Uncertainties and errors +### Random and systematic errors + +| Random error | Systematic error | +| --- | --- | +| Caused by imperfect measurements and is present in every measurement. | Caused by a flaw in experiment design or in the procedure. | +| Can be reduced (but not avoided) by repeated trials or measurements. | Cannot be reduced by repeated measurements, but can be avoided completely. | +| Error in precision. | Error in accuracy. | + +!!! example + - The failure to account for fluid evaporating at high temperatures is a systematic error, as it cannot be minimised by repeated measurements. + - The addition of slightly more solute due to uncertainty in instrument data is a random error, as it can be reduced by averaging the result of multiple trials. + +(Source: Kognity) + +### Uncertainties + +Uncertainties are stated in the form of [value] ± [uncertainty]. A value is only as precise as its absolute uncertainty. Absolute uncertainty of **measurement** is usually represented to only 1 significant digit. + +!!! note + Variables with uncertainty use an uppercase delta for their uncertainty value: $a ± \Delta a$ + + - The absolute uncertainty of a number is written in the same unit as the value. + - The percentage uncertainty of a number is the written as a percentage of the value. + +!!! example + - Absolute uncertainty: 1.0 g ± 0.1 g + - Percentage uncertainty: 1.0 g ± 10% + +To determine a measurement's absolute uncertainty, if: + + - the instrument states its uncertainty, use that. + - an analog instrument is used, the last digit is estimated and appended to the end of the reported value. The estimated digit is uncertain by 5 at its order of magnitude. + - a digital instrument is used, the last reported digit is uncertain by 1 at its order of magnitude. + +!!! example + - A ruler has millimetre markings. A pencil placed alongside the ruler has its tip just past 14 mm but before 15 mm. The pencil is 14.5 mm ± 0.5 mm long. + - A digital scale reads 0.66 kg for the mass of a human body. The human body has a mass of 0.66 kg ± 0.01 kg. + +!!! info + See [Dealing with Uncertainties](/resources/g11/physics-uncertainties.pdf) for how to perform operations with uncertainties. + +### Error bars + +Error bars represent the uncertainty of the data, typically representing that data point's standard deviation, and can be both horizontal or vertical. + +(Source: Kognity) + +!!! note + On a graph, a data point with uncertain values is written as $(x ± \Delta x, y ± \Delta y)$ + +### Uncertainty of gradient and intercepts + +!!! note "Definition" + - The **line of best fit** is the line that passes through **all error bars** while passing as closely as possible to all data points. + - The **minimum and maximum lines** are lines that minimise/maximise their slopes while still passing through **all error bars.** + +!!! warning + - Use solid lines for lines representing **continuous data** and dotted lines for **discrete data**. + +(Source: Kognity) + +The uncertainty of the **slope** of the line of best fit is the difference between the maximum and minimum slopes. +$$m_{best fit} ± m_{max}-m_{min}$$ +The uncertainty of the **intercepts** is the difference between the intercepts of the maximum and minimum lines. +$$intercept_{best fit} ± intercept_{max} - intercept_{min}$$ + ## 1.3 - Vectors and scalars @@ -89,3 +155,4 @@ The order of magnitude of a number can be found by converting it to scientific n - [IB SL Physics Syllabus](/resources/g11/ib-physics-syllabus.pdf) - [Dealing with Uncertainties](/resources/g11/physics-uncertainties.pdf) - [Linearising Data](/resources/g11/linearising-data.pdf) + - [External: IB Physics Notes](https://ibphysics.org)