238 lines
10 KiB
Markdown
238 lines
10 KiB
Markdown
# SL Physics - A
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The course code for this page is **SPH3U7**.
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## 1.1 - Measurements in physics
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### Fundamental units
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Every other SI unit is derived from the fundamental SI units. Memorise these!
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| Quantity type | Unit | Symbol |
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| --- | --- | --- |
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| Time | Second | s |
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| Distance | Metre | m |
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| Mass | Kilogram | kg |
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| Electric current | Ampere | A |
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| Temperature | Kelvin | K |
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| Amount of substance | Mole | mol |
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| Luminous intensity | Candela | cd |
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### Metric prefixes
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Every SI unit can be expanded with metric prefixes.
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!!! example
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milli + metre = millimetre ($10^{-3}$) m
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| Prefix | Abbreviation | Value | Inverse ($10^{-n}$) abbreviation | Inverse prefix |
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| --- | --- | --- | --- | --- |
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| deca- | da | $10^1$ | d | deci- |
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| hecto- | h | $10^2$ | c | centi- |
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| kilo- | k | $10^3$ | m | milli- |
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| mega- | M | $10^6$ | µ | micro- |
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| giga- | G | $10^9$ | n | nano- |
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| tera- | T | $10^{12}$ | p | pico- |
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| peta- | P | $10^{15}$ | f | femto- |
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| exa- | E | $10^{18}$ | a | atto- |
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### Significant figures
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- The leftmost non-zero digit is the **most significant digit**.
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- If there is no decimal point, the rightmost non-zero digit is the **least significant digit**.
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- Otherwise, the right-most digit (including zeroes) is the least significant digit.
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- All digits between the most and least significant digits are significant.
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- Pure (discrete) numbers are unitless and have infinite significant figures.
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!!! example
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In $123000$, there are 3 significant digits.<br>
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In $0.1230$, there are 4 significant digits.
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- When adding or subtracting significant figures, the answer has the **same number of decimals** as the number with the lowest number of decimal points.
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- 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.
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- Values of a calculated result can be **no more precise** than the least precise value used.
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!!! example
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$$1.25 + 1.20 = 2.45$$
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$$1.24 + 1.2 = 2.4$$
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$$1.2 × 2 = 2$$
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$$1.2 × 2.0 = 2.4$$
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!!! warning
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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**.
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$$1.25 + 1.2 = 2.4$$
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$$1.35 + 1.2 = 2.6$$
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### Scientific notation
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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.<br>
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!!! example
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The speed of light is 300 000 000 ms<sup>-1</sup>, or $3×10^8$ ms<sup>-1</sup>.
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### Orders of magnitude
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The order of magnitude of a number can be found by converting it to scientific notation and taking its power of 10.
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!!! example
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- The order of magnitude of $212000$, or $2.12×10^{5}$, is 5.
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- The order of magnitude of $0.212$, or $2.12×10^{-1}$, is -1.
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## 1.2 - Uncertainties and errors
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### Random and systematic errors
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| Random error | Systematic error |
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| --- | --- |
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| Caused by imperfect measurements and is present in every measurement. | Caused by a flaw in experiment design or in the procedure. |
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| Can be reduced (but not avoided) by repeated trials or measurements. | Cannot be reduced by repeated measurements, but can be avoided completely. |
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| Error in precision. | Error in accuracy. |
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!!! example
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- The failure to account for fluid evaporating at high temperatures is a systematic error, as it cannot be minimised by repeated measurements.
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- 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.
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<img src="/resources/images/types-of-error.png" width=700>(Source: Kognity)</img>
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### Uncertainties
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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.
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!!! note
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Variables with uncertainty use an uppercase delta for their uncertainty value: $a ± \Delta a$
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- The absolute uncertainty of a number is written in the same unit as the value.
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- The percentage uncertainty of a number is the written as a percentage of the value.
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!!! example
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- Absolute uncertainty: 1.0 g ± 0.1 g
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- Percentage uncertainty: 1.0 g ± 10%
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To determine a measurement's absolute uncertainty, if:
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- the instrument states its uncertainty, use that.
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- 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.
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- a digital instrument is used, the last reported digit is uncertain by 1 at its order of magnitude.
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!!! example
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- 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.
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- 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.
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See [Dealing with Uncertainties](/resources/g11/physics-uncertainties.pdf) for how to perform **operations with uncertainties**.
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### Error bars
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Error bars represent the uncertainty of the data, typically representing that data point's standard deviation, and can be both horizontal or vertical.
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<img src="/resources/images/error-bars.png" width=600>(Source: Kognity)</img>
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!!! note
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On a graph, a data point with uncertain values is written as $(x ± \Delta x, y ± \Delta y)$
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### Uncertainty of gradient and intercepts
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!!! note "Definition"
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- 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.
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- The **minimum and maximum lines** are lines that minimise/maximise their slopes while passing through the first and last **error bars**.
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!!! warning
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- Use solid lines for lines representing **continuous data** and dotted lines for **discrete data**.
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<img src="/resources/images/error-slopes.png" width=700>(Source: Kognity)</img>
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The uncertainty of the **slope** of the line of best fit is the difference between the maximum and minimum slopes.
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$$m_{best fit} ± m_{max}-m_{min}$$
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The uncertainty of the **intercepts** is the difference between the intercepts of the maximum and minimum lines.
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$$intercept_{best fit} ± intercept_{max} - intercept_{min}$$
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## 1.3 - Vectors and scalars
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!!! note "Definition"
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- **Scalar:** A physical quantity with a numerical value (magnitude) and a unit.
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- **Vector:** A physical quantity with a numerical value (magnitude), a unit, and a **direction.**
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??? example
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- Scalar quantities include speed, distance, mass, temperature, pressure, time, frequency, current, voltage, and more.
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- Vector quantities include velocity, displacement, acceleration, force (e.g., weight), momentum, impulse, and more.
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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.
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- The **standard form** of a vector is expressed as its magnitude followed by its unit followed by its direction in square brackets.
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$$\vec{a} = 1\text{ m }[N 45° E]$$
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- The **component form** of a vector is expressed as the location of its head on a cartesian plane if its tail were at $(0, 0)$.
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$$\vec{a} = (1, 1)$$
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- The **magnitude** of a vector can be expressed as the absolute value of a vector.
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$$|\vec{a}| = 1 \text{ m}$$
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### Adding/subtracting vectors diagrammatically
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1. Draw the first vector.
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2. Draw the second vector with its *tail* at the *head* of the first vector.
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3. Repeat step 2 as necessary for as many vectors as you want by attaching them to the *head* of the last vector.
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4. Draw a new (**resultant**) vector from the *tail* of the first vector to the *head* of the last vector.
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<img src="/resources/images/vector-add-direction.png" width=700>(Source: Kognity)</img>
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When subtracting a vector, **negate** the vector being subtracted by giving it an opposite direction and then add the vectors.
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<img src="/resources/images/vector-subtract-direction.png" width=700>(Source: Kognity)</img>
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### Adding/subtracting vectors algebraically
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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.
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$$\vec{F}_x + \vec{F}_y = \vec{F}$$
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!!! info "Reminder"
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The **component form** of a vector is expressed as $(|\vec{a}_x|, |\vec{a}_y|)$
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<img src="/resources/images/vector-simple-adding.png" width=700>(Source: Kognity)</img>
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By using the primary trignometric identities:
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$$
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|\vec{a}_{x}| = |\vec{a}|\cos\theta_{a} \\
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|\vec{a}_{y}| = |\vec{a}|\sin\theta_{a}
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$$
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<img src="/resources/images/vector-decomposition.png" width=700>(Source: Kognity)</img>
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Using their component forms, to:
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- add two vectors, add their x- and y-coordinates together.
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- subtract two vectors, subtract their x- and y-coordinates together.
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$$
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(a_{x}, a_{y}) + (b_{x}, b_{y}) = (a_{x} + b_{x}, a_{y} + b_{y}) \\
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(a_{x}, a_{y}) - (b_{x}, b_{y}) = (a_{x} - b_{x}, a_{y} - b_{y})
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$$
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The length of resultant vector can then be found using the Pythagorean theorem.
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$$
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|\vec{c}|=\sqrt{c_{x}^2 + c_{y}^2}
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$$
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To find the resultant direction, use inverse tan to calculate the angle of the vector using the lengths of its components.
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$$
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\vec{c}_{direction} = \tan^{-1} \frac{c_y}{c_x}
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$$
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### Multiplying vectors and scalars
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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.
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$$\vec{v} × s = (|\vec{v}|×s)[\theta_{v}]$$
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!!! example
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$$3 \text{ m} · 47 \text{ ms}^{-1}[N20°E] = 141 \text{ ms}^{-1}[N20°E]$$
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## Resources
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- [IB Physics Data Booklet](/resources/g11/ib-physics-data-booklet.pdf)
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- [IB SL Physics Syllabus](/resources/g11/ib-physics-syllabus.pdf)
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- [Dealing with Uncertainties](/resources/g11/physics-uncertainties.pdf)
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- [Linearising Data](/resources/g11/linearising-data.pdf)
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- [External: IB Physics Notes](https://ibphysics.org)
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