# SL Physics - A The course code for this page is **SPH3U7**. ## 1.1 - Measurements in physics ### Fundamental units Every other SI unit is derived from the fundamental SI units. Memorise these! | Quantity type | Unit | Symbol | | --- | --- | --- | | Time | Second | s | | Distance | Metre | m | | Mass | Kilogram | kg | | Electric current | Ampere | A | | Temperature | Kelvin | K | | Amount of substance | Mole | mol | | Luminous intensity | Candela | cd | ### Metric prefixes Every SI unit can be expanded with metric prefixes. !!! example milli + metre = millimetre ($10^{-3}$) m | Prefix | Abbreviation | Value | Inverse ($10^{-n}$) abbreviation | Inverse prefix | | --- | --- | --- | --- | --- | | deca- | da | $10^1$ | d | deci- | | hecto- | h | $10^2$ | c | centi- | | kilo- | k | $10^3$ | m | milli- | | mega- | M | $10^6$ | µ | micro- | | giga- | G | $10^9$ | n | nano- | | tera- | T | $10^{12}$ | p | pico- | | peta- | P | $10^{15}$ | f | femto- | | exa- | E | $10^{18}$ | a | atto- | ### Significant figures - The leftmost non-zero digit is the **most significant digit**. - If there is no decimal point, the rightmost non-zero digit is the **least significant digit**. - Otherwise, the right-most digit (including zeroes) is the least significant digit. - All digits between the most and least significant digits are significant. - Pure (discrete) numbers are unitless and have infinite significant figures. !!! example In $123000$, there are 3 significant digits.
In $0.1230$, there are 4 significant digits. - When adding or subtracting significant figures, the answer has the **same number of decimals** as the number with the lowest number of decimal points. - When multiplying or dividing significant figures, the answer has the **same number of significant figures** as the number with the lowest number of significant figures. - Values of a calculated result can be **no more precise** than the least precise value used. !!! example $$1.25 + 1.20 = 2.45$$ $$1.24 + 1.2 = 2.4$$ $$1.2 × 2 = 2$$ $$1.2 × 2.0 = 2.4$$ !!! 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**. $$1.25 + 1.2 = 2.4$$ $$1.35 + 1.2 = 2.6$$ ### Scientific notation Scientific notation is written in the form of $m×10^{n}$, where $1 \leq m < 10, n \in \mathbb{Z}$. All digits before the multiplication sign in scientific notation are significant.
!!! example The speed of light is 300 000 000 ms-1, or $3×10^8$ ms-1. ### Orders of magnitude 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. ## 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. 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 **as many error bars as possible** while passing as closely as possible to all data points. - The **minimum and maximum lines** are lines that minimise/maximise their slopes while passing through the first and last **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 !!! note "Definition" - **Scalar:** A physical quantity with a numerical value (magnitude) and a unit. - **Vector:** A physical quantity with a numerical value (magnitude), a unit, and a **direction.** ??? example - Scalar quantities include speed, distance, mass, temperature, pressure, time, frequency, current, voltage, and more. - Vector quantities include velocity, displacement, acceleration, force (e.g., weight), momentum, impulse, and more. Vectors are drawn as arrows whose length represents their scale/magnitude and their orientation refer to their direction. A variable representing a vector is written with a right-pointing arrow above it. - The **standard form** of a vector is expressed as its magnitude followed by its unit followed by its direction in square brackets. $$\vec{a} = 1\text{ m }[N 45° E]$$ - The **component form** of a vector is expressed as the location of its head on a cartesian plane if its tail were at $(0, 0)$. $$\vec{a} = (1, 1)$$ - The **magnitude** of a vector can be expressed as the absolute value of a vector. $$|\vec{a}| = 1 \text{ m}$$ ### Adding/subtracting vectors diagrammatically 1. Draw the first vector. 2. Draw the second vector with its *tail* at the *head* of the first vector. 3. Repeat step 2 as necessary for as many vectors as you want by attaching them to the *head* of the last vector. 4. Draw a new (**resultant**) vector from the *tail* of the first vector to the *head* of the last vector. (Source: Kognity) When subtracting a vector, **negate** the vector being subtracted by giving it an opposite direction and then add the vectors. (Source: Kognity) ### Adding/subtracting vectors algebraically Vectors can be broken up into two vectors (**"components"**) laying on the x- and y-axes via trigonometry such that the resultant of the two components is the original vector. This is especially helpful when adding larger (3+) numbers of vectors. $$\vec{F}_x + \vec{F}_y = \vec{F}$$ !!! info "Reminder" The **component form** of a vector is expressed as $(|\vec{a}_x|, |\vec{a}_y|)$ (Source: Kognity) By using the primary trignometric identities: $$ |\vec{a}_{x}| = |\vec{a}|\cos\theta_{a} \\ |\vec{a}_{y}| = |\vec{a}|\sin\theta_{a} $$ (Source: Kognity) Using their component forms, to: - add two vectors, add their x- and y-coordinates together. - subtract two vectors, subtract their x- and y-coordinates together. $$ (a_{x}, a_{y}) + (b_{x}, b_{y}) = (a_{x} + b_{x}, a_{y} + b_{y}) \\ (a_{x}, a_{y}) - (b_{x}, b_{y}) = (a_{x} - b_{x}, a_{y} - b_{y}) $$ The length of resultant vector can then be found using the Pythagorean theorem. $$ |\vec{c}|=\sqrt{c_{x}^2 + c_{y}^2} $$ To find the resultant direction, use inverse tan to calculate the angle of the vector using the lengths of its components. $$ \vec{c}_{direction} = \tan^{-1} \frac{c_y}{c_x} $$ ### Multiplying vectors and scalars The product of a vector multiplied by a scalar is a vector with a magnitude of the vector multiplied by the scalar with the same direction as the original vector. $$\vec{v} × s = (|\vec{v}|×s)[\theta_{v}]$$ !!! example $$3 \text{ m} · 47 \text{ ms}^{-1}[N20°E] = 141 \text{ ms}^{-1}[N20°E]$$ ## Resources - [IB Physics Data Booklet](/resources/g11/ib-physics-data-booklet.pdf) - [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)