Physics skills
The skills you use in every topic, every practical and the Internal Assessment: units, significant figures, uncertainties, graphs and vectors. In the IB guide they are part of the "Tools" (especially Tool 3: Mathematics). Exam questions test them throughout Papers 1 and 2, so it's worth getting them right early.
Knowledge and science
Nature of science
Measurement. Every measurement has an uncertainty. A result without one can't be compared fairly with a prediction or with anyone else's result.
Evidence. Two results only disagree if their uncertainty ranges don't overlap. Uncertainties tell scientists whether a surprising result is real evidence or just a measurement that happened to come out high.
Shared endeavour. The SI units are agreed internationally. Since 2019, all seven base units have been defined from fixed constants of nature, such as the speed of light and the Planck constant, so any well-equipped laboratory in the world can reproduce them.
ToK: questions to think about
- Can we ever know the "true value"? We only ever have measurements with uncertainties. Is the true value something we discover, or an idea we use to make sense of our data?
- How certain is certain enough? A bridge engineer and a particle physicist need very different levels of certainty. Who decides what counts as "precise enough"?
- Are units invented or discovered? The metre was once a metal bar in Paris and is now defined using the speed of light. Did the metre change?
- Why draw graphs? A best-fit line asks us to ignore some of the scatter in our data. When is it right to trust the pattern over the individual points?
1. SI units, prefixes and scientific notation
Physics uses the International System of Units (SI). Every other unit can be built from the seven base units:
| Quantity | Unit | Symbol |
|---|---|---|
| mass | kilogram | kg |
| length | metre | m |
| time | second | s |
| electric current | ampere | A |
| temperature | kelvin | K |
| amount of substance | mole | mol |
| luminous intensity | candela | cd |
Derived units are combinations of base units. For example, the newton comes from $F = ma$: $1\ \text{N} = 1\ \text{kg m s}^{-2}$. The joule comes from $W = Fs$: $1\ \text{J} = 1\ \text{N m} = 1\ \text{kg m}^{2}\,\text{s}^{-2}$. Checking that both sides of an equation have the same units is a quick way to catch mistakes.
Prefixes (given in the data booklet) make very large and very small numbers easier to handle:
| Prefix | Symbol | Factor |
|---|---|---|
| tera | T | $10^{12}$ |
| giga | G | $10^{9}$ |
| mega | M | $10^{6}$ |
| kilo | k | $10^{3}$ |
| centi | c | $10^{-2}$ |
| milli | m | $10^{-3}$ |
| micro | µ | $10^{-6}$ |
| nano | n | $10^{-9}$ |
| pico | p | $10^{-12}$ |
Scientific notation writes a number as $a \times 10^{n}$ with $1 \le a < 10$. For example, $0.000\,47 = 4.7\times10^{-4}$ and $93\,000\,000 = 9.3\times10^{7}$. It also makes the number of significant figures clear (section 3).
Orders of magnitude: the power of ten closest to a value. A person's mass (about 70 kg) is of order $10^{2}$ kg; the diameter of an atom is of order $10^{-10}$ m. Comparing orders of magnitude is a quick way to check whether an answer is sensible.
2. Accuracy, precision and errors
- A measurement is accurate if it is close to the true (or accepted) value.
- A set of measurements is precise if the readings are close to each other, whether or not they are close to the true value.
Random errors
- Scatter readings above and below the true value, by a different amount each time. They reduce precision.
- Examples: reaction time with a stopwatch; reading a meter whose value flickers; small changes in conditions.
- Reduce them by repeating readings and taking the mean, or by using a more precise instrument.
Systematic errors
- Shift every reading in the same direction, by the same amount (or the same fraction). They reduce accuracy.
- Examples: a zero error (a balance reading 0.2 g with nothing on it); a stretched tape measure; heat lost to the surroundings in every trial.
- Reduce them by zeroing or recalibrating the instrument, or by improving the method. Repeating doesn't help.
On a graph, a systematic error often shows up as an unexpected intercept: a straight line that should pass through the origin but doesn't. Random errors show up as scatter of the points about the line.
3. Significant figures
Counting significant figures
- Non-zero digits always count: 472 has 3 s.f.
- Zeros between non-zero digits count: 5.006 has 4 s.f.
- Zeros at the start of a number never count: 0.0420 has 3 s.f. (the 4, the 2 and the final 0).
- Zeros at the end of a decimal count: 2.50 has 3 s.f. They show the measurement was made to the nearest 0.01.
- Zeros at the end of a whole number are ambiguous: 3600 could have 2, 3 or 4 s.f. Scientific notation removes the doubt: $3.6\times10^{3}$ (2 s.f.) or $3.600\times10^{3}$ (4 s.f.).
Significant figures in calculations
Multiplying, dividing, powers and roots: give the answer to the same number of significant figures as the least precise value used. $4.8 \times 3.162 = 15.1776$, which rounds to $15$ (2 s.f.).
Adding and subtracting: give the answer to the same number of decimal places as the value with the fewest. $12.45 + 3.1 = 15.55$, which rounds to $15.6$ (1 d.p.).
Keep extra digits in your calculator during a calculation, and round only at the end. Rounding at every step builds up errors. A long string of digits copied from the calculator claims a precision your data doesn't have.
4. Recording uncertainties
Every measurement is written as a best value ± absolute uncertainty, with a unit: for example $l = (24.5 \pm 0.1)$ cm.
- Analogue scale (a ruler or a thermometer): the uncertainty is usually half the smallest division, so ± 0.5 mm for a millimetre ruler. Measuring a length with a ruler involves reading both ends, so some teachers use ± 1 mm.
- Digital instrument: the uncertainty is at least ± 1 in the last digit, so ± 0.01 g for a balance reading 12.34 g.
- Repeated readings: use the mean, $\bar{x}$, as the best value and half the range as the uncertainty: $\Delta x = \dfrac{x_{\max} - x_{\min}}{2}$. If this is smaller than the instrument's uncertainty, use the instrument's uncertainty instead.
- Sometimes the method matters more than the instrument. A stopwatch reads to 0.01 s, but human reaction time adds an uncertainty of about 0.1–0.2 s.
Worked example: repeated timings
Five students time the same falling ball: 2.31 s, 2.45 s, 2.38 s, 2.29 s and 2.42 s. Give the result with its uncertainty.
Mean: the five times add up to $11.85$ s, so $\bar{t} = \dfrac{11.85}{5} = 2.37$ s.
Half the range: $\Delta t = \dfrac{2.45 - 2.29}{2} = 0.08$ s. So $t = (2.37 \pm 0.08)$ s.
The same uncertainty can be written three ways. For $t = (2.37 \pm 0.08)$ s:
- absolute uncertainty: $\Delta t = 0.08$ s (same unit as the value);
- fractional uncertainty: $\dfrac{\Delta t}{t} = \dfrac{0.08}{2.37} = 0.034$ (no unit);
- percentage uncertainty: $3.4\%$.
Quoting uncertainties: round the absolute uncertainty to 1 significant figure (2 is sometimes acceptable), and give the best value to the same decimal place. Write $(9.7 \pm 0.3)\ \text{m s}^{-2}$, not $(9.748 \pm 0.278)\ \text{m s}^{-2}$ or $(9.748 \pm 0.3)\ \text{m s}^{-2}$.
Percentage error is different from percentage uncertainty. It compares your result with an accepted value: $\dfrac{|\text{experimental} - \text{accepted}|}{\text{accepted}} \times 100\%$. If the percentage error is bigger than the percentage uncertainty, there is probably a systematic error.
5. Propagating uncertainties
When you calculate a result from measured values, their uncertainties carry through to the result. The data booklet gives three rules:
Adding or subtracting: add the absolute uncertainties.
$$y = a \pm b \quad\Rightarrow\quad \Delta y = \Delta a + \Delta b$$Multiplying or dividing: add the fractional (or percentage) uncertainties.
$$y = \frac{ab}{c} \quad\Rightarrow\quad \frac{\Delta y}{y} = \frac{\Delta a}{a} + \frac{\Delta b}{b} + \frac{\Delta c}{c}$$Powers: multiply the fractional uncertainty by the size of the power.
$$y = a^{n} \quad\Rightarrow\quad \frac{\Delta y}{y} = \left|n\,\frac{\Delta a}{a}\right|$$- Uncertainties always add, even when you subtract the quantities. Subtracting two similar values can give a result with a very large percentage uncertainty.
- Exact numbers, such as the 2 in $2\pi r$ or the $\frac{4}{3}$ in $\frac{4}{3}\pi r^3$, have no uncertainty, so they don't appear in the uncertainty calculation.
- A square root is a power of $\frac{1}{2}$, so it halves the fractional uncertainty.
Worked example: a temperature rise (subtracting)
Water starts at $(21.5 \pm 0.5)$ °C and ends at $(34.0 \pm 0.5)$ °C. Find the temperature rise.
$\Delta T = 34.0 - 21.5 = 12.5$ °C, with uncertainty $0.5 + 0.5 = 1.0$ °C. So the rise is $(12.5 \pm 1.0)$ °C, which is an 8% uncertainty, even though each reading was only about 2% uncertain.
Worked example: density (dividing)
A stone has mass $(240 \pm 2)$ g and volume $(88 \pm 3)\ \text{cm}^3$. Find its density.
$\rho = \dfrac{m}{V} = \dfrac{240}{88} = 2.73\ \text{g cm}^{-3}$.
$\dfrac{\Delta\rho}{\rho} = \dfrac{2}{240} + \dfrac{3}{88} = 0.008 + 0.034 = 0.042$, so $\Delta\rho = 0.042 \times 2.73 = 0.1\ \text{g cm}^{-3}$.
$\rho = (2.7 \pm 0.1)\ \text{g cm}^{-3}$. Notice that the volume uncertainty dominates: improving the volume measurement would help much more than a better balance.
Worked example: g from a pendulum (powers)
A pendulum has length $l = (0.800 \pm 0.005)$ m and period $T = (1.80 \pm 0.02)$ s. Find $g$ using $g = \dfrac{4\pi^2 l}{T^2}$.
$g = \dfrac{4\pi^2 \times 0.800}{1.80^2} = 9.75\ \text{m s}^{-2}$.
$\dfrac{\Delta g}{g} = \dfrac{\Delta l}{l} + 2\,\dfrac{\Delta T}{T} = \dfrac{0.005}{0.800} + 2 \times \dfrac{0.02}{1.80} = 0.006 + 0.022 = 0.028$. The $T$ term is doubled because $T$ is squared; $4\pi^2$ is exact.
$\Delta g = 0.028 \times 9.75 = 0.3\ \text{m s}^{-2}$, so $g = (9.7 \pm 0.3)\ \text{m s}^{-2}$. The accepted value, 9.8, lies inside this range, so the result agrees with it.
6. Graphs and best-fit lines
- Put the independent variable (the one you change) on the $x$-axis and the dependent variable (the one you measure) on the $y$-axis.
- Label each axis with the quantity and its unit, for example "extension / cm".
- Choose scales so that the points fill most of the graph area, and use easy steps (1, 2 or 5 per division).
- Draw uncertainty bars on each point, showing $\pm\Delta y$ (and $\pm\Delta x$ if it is big enough to see).
- Draw a single smooth line or curve of best fit, with the points scattered evenly either side. It should pass through as many uncertainty bars as possible. Don't join the dots, and don't force it through the origin.
- Find the gradient from a large triangle drawn on the line, using two points far apart. Don't use two data points.
Linearising: making a straight line
Straight-line graphs are the easiest to analyse. Compare your equation with $y = mx + c$ and choose what to plot so that you get a straight line. The gradient and intercept then give physical quantities.
| Relationship | Plot | Gradient |
|---|---|---|
| $F = kx$ (spring) | $F$ against $x$ | $k$ |
| $s = \frac{1}{2}gt^2$ (falling from rest) | $s$ against $t^2$ | $\frac{g}{2}$ |
| $T = 2\pi\sqrt{\frac{l}{g}}$ (pendulum) | $T^2$ against $l$ | $\frac{4\pi^2}{g}$ |
| $v = u + at$ | $v$ against $t$ | $a$ (intercept $u$) |
In the first three, the line should pass through the origin. An intercept that should be zero but isn't usually means a systematic error (such as a zero error) or a missing effect (such as friction). That's useful information, not a failure.
7. Uncertainty in the gradient and intercept
To find the uncertainty in a gradient:
- Draw the line of best fit, and find its gradient and intercept.
- Draw the steepest line that still passes through every uncertainty bar, and the shallowest such line. These are the lines of maximum and minimum gradient.
- Find the gradient and intercept of each.
- The uncertainty is half the difference: $\Delta m = \dfrac{m_{\max} - m_{\min}}{2}$ and $\Delta c = \dfrac{c_{\max} - c_{\min}}{2}$.
Worked example: the spring constant
From the graph above, the best-fit line has gradient 0.51 N cm⁻¹ and an intercept of about 0. The maximum-gradient line has gradient 0.55 N cm⁻¹ and intercept −0.13 N. The minimum-gradient line has gradient 0.46 N cm⁻¹ and intercept +0.16 N.
$\Delta k = \dfrac{0.55 - 0.46}{2} = 0.05\ \text{N cm}^{-1}$, so $k = (0.51 \pm 0.05)\ \text{N cm}^{-1} = (51 \pm 5)\ \text{N m}^{-1}$.
Intercept: $\Delta c = \dfrac{0.16 - (-0.13)}{2} = 0.1$ N, so $c = (0.0 \pm 0.1)$ N. The intercept is consistent with zero, as Hooke's law predicts. There's no sign of a zero error.
8. Vectors
A scalar has size (magnitude) only. A vector has magnitude and direction.
| Vectors | Scalars |
|---|---|
| displacement, velocity, acceleration | distance, speed, time |
| force (including weight), momentum, impulse | mass, energy, work, power |
| electric, magnetic and gravitational field strength | temperature, density, pressure, electric potential |
A vector is drawn as an arrow: its length shows the magnitude and the arrowhead shows the direction. Two vectors are equal if they have the same magnitude and direction, wherever they are drawn. Multiplying a vector by a scalar changes its length; multiplying by a negative number also reverses it.
Adding and subtracting vectors
For two perpendicular vectors, use Pythagoras for the magnitude and trigonometry for the direction. For other angles, draw an accurate scale diagram, or resolve each vector into components (below), add the components, and recombine.
Resolving a vector into components
Any vector can be split into two perpendicular components. For a vector $A$ at angle $\theta$ to the $x$-axis:
and to rebuild the vector from its components:
$$A = \sqrt{A_x^2 + A_y^2} \qquad \tan\theta = \frac{A_y}{A_x}$$Signs matter. A component pointing left or down is negative. When you rebuild a vector, find the angle from $\tan\theta = \left|\frac{A_y}{A_x}\right|$, then use a quick sketch to decide which quadrant it points into. Your calculator can't tell the difference between "left and down" and "right and up".
Worked example: components and back again
(a) A rope pulls on a crate with a force of 25 N at 40° above the horizontal. Find the horizontal and vertical components.
$F_x = 25\cos 40° = 19$ N (horizontal), $F_y = 25\sin 40° = 16$ N (vertical).
(b) A force has components $F_x = -3.0$ N and $F_y = +4.0$ N. Find its magnitude and direction.
$F = \sqrt{3.0^2 + 4.0^2} = 5.0$ N. $\tan\theta = \dfrac{4.0}{3.0}$, so $\theta = 53°$.
$F_x$ is negative and $F_y$ is positive, so the force points up and to the left: 53° above the negative $x$-axis.
9. Common mistakes
- Leaving out units, or the uncertainty, from a final answer.
- Copying every digit from the calculator. Round the uncertainty to 1 s.f. and match the value to it.
- Subtracting uncertainties when the quantities are subtracted. Absolute uncertainties always add.
- Adding absolute uncertainties when multiplying or dividing. Use fractional or percentage uncertainties instead.
- Forgetting to multiply by the power, for example doubling the fractional uncertainty for $T^2$.
- Thinking repeats remove a systematic error. They only reduce random errors.
- Finding a gradient from two data points rather than from the best-fit line, or using a small triangle.
- Forcing a best-fit line through the origin. Let the data decide; the intercept is information.
- Mixing up accuracy and precision. Accurate means close to the true value; precise means close together.
- Losing the sign of a vector component, or having the calculator in the wrong angle mode.
10. Check your understanding
How many significant figures are there in 0.003 050?
Four: 3, 0, 5 and the final 0. The leading zeros don't count. In scientific notation it's $3.050\times10^{-3}$.
A length is measured as (50.0 ± 0.2) cm. A second length is (48.5 ± 0.2) cm. What is the difference between them, with its percentage uncertainty?
$50.0 - 48.5 = 1.5$ cm, with uncertainty $0.2 + 0.2 = 0.4$ cm. That's a percentage uncertainty of $\frac{0.4}{1.5} = 27\%$, even though each length was measured to 0.4%. Subtracting similar values magnifies the percentage uncertainty.
The radius of a circle has an uncertainty of 3%. What is the percentage uncertainty in its area?
$A = \pi r^2$, so the fractional uncertainty doubles: 6%. $\pi$ is exact.
Every reading on a newton-meter is 0.3 N too high. Is this a random or a systematic error, and will repeating the measurements help?
Systematic (a zero error). Repeating won't help, because every reading is shifted the same way. Zero the meter before use, or subtract 0.3 N from every reading.
A vector has components $A_x = -6.0$ and $A_y = -8.0$. Find its magnitude, and describe its direction.
$A = \sqrt{6.0^2 + 8.0^2} = 10$. $\tan\theta = \frac{8.0}{6.0}$, so $\theta = 53°$. Both components are negative, so it points left and down: 53° below the negative $x$-axis.