10 KiB
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 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
- Draw the first vector.
- Draw the second vector with its tail at the head of the first vector.
- Repeat step 2 as necessary for as many vectors as you want by attaching them to the head of the last vector.
- 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 \[3 \text{ m} · 47 \text{ ms}^{-1}[N20°E] = 141 \text{ ms}^{-1}[N20°E]\]