# 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 **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 !!! note "Definition" - **Scalar:** A physical quantity with a numerical value and unit. - **Vector:** A physical quantity with a numerical value, unit, and **direction.** ??? example - Physical quantities represented by scalars include speed, distance, mass, temperature, pressure, time, frequency, current, voltage, and more. - Physical quantities represented by vectors include velocity, displacement, acceleration, force (e.g., weight), momentum, impulse, and more. Vectors are represented as arrows whose length represents their scale/magnitude and their orientation refer to their direction. ### 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 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. (Source: Kognity) ### Parallelogram rule The parallelogram rule states that the sum of two vectors that form two sides of a parallelogram is the diagonal of that parallelogram. (Source: Kognity) ### 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. $$v[direction] × s = (v×s)[direction]$$ ### Vector decomposition By breaking up a vector into lengths along the x- and y-axes, the sum of two vectors can be calculated algebraically. (Source: Kognity) For vector $\textbf{a}$ and vector $\textbf{b}$: $$ a_{x} = a\cos\theta_{a} \\ a_{y} = a\sin\theta_{a} $$ Proof: $a=\sqrt{a^{2}_{x}+a^{2}_{y}} \\$ $=\sqrt{(a\cos\theta_{a})^2 + (a\sin\theta{a})^2} \\$ $=\sqrt{a^2(\cos\theta_{a}^2 + \sin\theta_{a}^2)} \\$ $=\sqrt{a^2} \\$ $=a$ From the diagram above, we can figure out that: $$ r_{magnitude}=\sqrt{(a\cos\theta_{a} + b\cos\theta_{b})^2 + (a\sin\theta_{a} + b\cos\theta_{b})^2} \\ r_{direction}=\tan^{-1}(\frac{a\sin\theta_{a} + b\sin\theta_{b}}{a\cos\theta_{a} + b\cos\theta_{b}}) $$ ## 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)