PHYSICS · LESSON 09 OF 9
Laboratory graphs and evidence
Plot measured data, draw the best straight line, and read physics from its slope and intercept — with units and a realistic uncertainty.
What this lesson explains
Laboratory work is part of PHY 101. Physical laws are tested and constants are measured by plotting data and fitting a line. A graph averages out random scatter, shows whether a model fits, and reveals mistakes that a single calculation hides. Engineers use the same skill to calibrate sensors and extract material properties.
Before you begin
Straight-line equation
y = mx + b; slope m = y2 − y1 / x2 − x1; intercept b where the line meets the y-axis.
Units and significant figures
See “Units and measurement”. A slope has the unit of y divided by the unit of x.
The idea, made visible
Put the quantity you control (the independent variable) on the horizontal axis and the one you measure in response (the dependent variable) on the vertical axis. Label both axes with the quantity and its unit, and choose scales that spread the points over most of the graph.
If the relationship is linear, y = mx + b, the points should scatter around a straight line. Draw the best-fit line: roughly as many points above as below, following the trend — not simply joining the first and last points, and not forced through every point. Real data always scatter.
The slope m = Δy / Δx has units (y-unit)/(x-unit) and usually means something physical: the slope of position against time is velocity, of velocity against time is acceleration, of force against extension is the spring constant. Calculate it from two points on the line, far apart, not from two data points.
The intercept b is the value of y when x = 0. A theory may predict it (often zero); a non-zero intercept can indicate a systematic error, like a zero offset in the instrument.
Many laws are not linear, but can be made linear. For a falling object, d = 1 / 2gt2: plot d against t2 (not t) to get a straight line with slope g/2. For a pendulum, T = 2π√L/g: plot T2 against L, slope 4π2/g. This is called linearising the data.
Every measurement has uncertainty. Random errors scatter the points and can be reduced by repeating and averaging; systematic errors shift all readings the same way and cannot be removed by averaging. Report results as value ± uncertainty with a unit, and compare with an accepted value by percentage error.
| extension x (m) | force F (N) | F/x (N/m) |
|---|---|---|
| 0.01 | 2.1 | 210 |
| 0.02 | 3.9 | 195 |
| 0.03 | 6.2 | 207 |
| 0.04 | 7.9 | 198 |
| 0.05 | 10.1 | 202 |
| 0.06 | 11.9 | 198 |
The individual ratios scatter (195–210 N/m); the graph’s slope averages them. Reported: k = 200 ± 10 N/m.
Key terms
- Independent / dependent variable
- The quantity you choose or control (x-axis) / the quantity that responds (y-axis).
- Best-fit line
- The straight line that best represents the trend of the data, balancing points above and below. (A calculator or spreadsheet computes the least-squares line.)
- Slope and intercept
- Slope: rate of change of y with x, with units. Intercept: value of y when x = 0.
- Linearisation
- Plotting transformed quantities (such as t² or T²) so that a non-linear law appears as a straight line.
- Random vs systematic error
- Random: unpredictable scatter in both directions. Systematic: consistent bias in one direction (miscalibration, zero offset, reaction time).
- Percentage error
- ∣measured − accepted∣ / ∣accepted∣ × 100%, for a nonzero accepted value.
The formulas and what they mean
| Symbol | Meaning | Unit |
|---|---|---|
| m | slope of the graph | y-unit ÷ x-unit |
| b | intercept | y-unit |
| k | spring constant (example slope) | N/m |
| ±δ | uncertainty of a value | same as the value |
Slope from the line
m = y2 − y1 / x2 − x1
Conditions and limits: Use two widely separated points read from the best-fit line, not raw data points.
Meaning of common slopes
x–t: v; v–t: a; F–x: k (Hooke’s law); d–t²: g/2; T²–L: 4π²/g
Conditions and limits: Each requires the corresponding model to hold (constant velocity, constant acceleration, elastic limit, small pendulum swings).
Area under a graph
area under v–t = displacement; area under F–x = work
Conditions and limits: Area in (y-unit × x-unit).
Mean and spread of repeated readings
x̄ = x1 + … + xn / n; uncertainty ≈ xmax − xmin / 2
Conditions and limits: A simple estimate for a few repeated readings; laboratory courses use the standard deviation.
Why it works: Linearising free-fall data
A ball is dropped from rest; the theory says d = 1 / 2gt2.
A graph of d against t is a curve (a parabola), so its slope is not constant.
Hard to read a constant from a curve.
Compare d = (g / 2)(t2) with y = mx: take y = d and x = t2.
Match the form of a straight line through the origin.
Plot d against t2; the slope is g/2, so g = 2 × slope.
Units: m/s², as required.
If the points on the d–t² graph lie on a straight line through the origin, the data support the model.
A first worked example
From data to a result
- Decide which variable is independent (x) and dependent (y); if the law is non-linear, choose transformed variables that make it linear.
- Draw axes with quantities and units; choose scales so the data fill most of the grid.
- Plot the points carefully; add error bars if uncertainties are known.
- Draw the best-fit straight line (or use least squares).
- Find the slope from two distant points on the line; include units. Note the intercept.
- Interpret: what physical quantity is the slope? Compare with the expected value (percentage error) and discuss sources of error.
Velocity from a position–time graph
Problem. A position–time graph is a straight line through (0 s, 1 m) and (2 s, 7 m). Find the velocity.
Slope = 7 − 1 / 2 − 0 = 6 m / 2 s.
Rise over run with units.
= 3 m/s.
Slope of x–t is velocity.
Result: v = 3 m/s (constant, since the graph is straight).
What it means: The intercept 1 m is the starting position.
A different case
Spring constant from a graph
Problem. Force against extension gives a best-fit line through (0.02 m, 4 N) and (0.06 m, 12 N). Find k.
Slope = 12 − 4 / 0.06 − 0.02 = 8 N / 0.04 m = 200 N/m.
Two points on the line.
Hooke’s law F = kx, so the slope is k.
Interpret the slope physically.
Result: k = 200 N/m.
What it means: A stiffer spring would give a steeper line.
More worked cases
Each case below uses a different skill. Every step and result is shown.
Acceleration and displacement from a velocity–time graph
Problem. A v–t graph is a straight line from (0 s, 2 m/s) to (4 s, 10 m/s). Find the acceleration and the displacement in the 4 s.
a = slope = 10 − 2 / 4 = 2 m/s2.
Slope of v–t.
Displacement = area under the line = trapezium = 2 + 10 / 2 × 4 = 24 m.
Area under v–t.
Result: a = 2 m/s2; displacement 24 m.
What it means: Check with Δx = v₀t + ½at² = 8 + 16 = 24 m.
g from linearised free-fall data
Problem. A plot of drop distance d against t2 gives a straight line through the origin with slope 4.85 m/s2. Find g and compare with 9.81 m/s2.
d = 1 / 2gt2, so slope = g/2.
Linearised model.
g = 2 × 4.85 = 9.70 m/s2.
Solve for g.
Percentage error = ∣9.70 − 9.81∣ / 9.81 × 100% = 1.1%.
Compare with the accepted value.
Result: g ≈ 9.70 m/s2, 1.1% below the accepted value.
What it means: A consistently low result may indicate a systematic error such as air resistance or a timing delay.
Averaging repeated readings
Problem. Five timings of 10 pendulum swings: 14.2, 14.5, 14.3, 14.6, 14.4 s. Find the period with an uncertainty.
Mean: 14.2 + 14.5 + 14.3 + 14.6 + 14.4 / 5 = 14.4 s for 10 swings.
Average reduces random error.
Half-range: 14.6 − 14.2 / 2 = 0.2 s.
Simple uncertainty estimate.
Period T = 1.44 ± 0.02 s.
Divide both by 10: timing many swings reduces the relative effect of reaction time.
Result: T = 1.44 ± 0.02 s.
What it means: Timing 10 swings rather than one makes the reaction-time error ten times smaller per period.
Common misunderstandings
Misunderstanding: Joining the dots, or forcing the line through the first and last point.
Correct idea: Draw one best-fit line through the trend of all the points.
Misunderstanding: Calculating the slope from two data points.
Correct idea: Use two points on the fitted line, far apart.
Misunderstanding: Slope without units.
Correct idea: The slope’s unit is y-unit ÷ x-unit, and it often is the physical result.
Misunderstanding: Plotting a curved relationship and reading a “slope”.
Correct idea: Linearise first (e.g. d against t²).
Misunderstanding: “Averaging removes all errors.”
Correct idea: Averaging reduces random errors only; systematic errors remain.
Keep in mind
- m = y2 − y1 / x2 − x1Slope from the line
- x–t: v; v–t: a; F–x: k (Hooke’s law); d–t²: g/2; T²–L: 4π²/gMeaning of common slopes
- area under v–t = displacement; area under F–x = workArea under a graph
- x̄ = x1 + … + xn / n; uncertainty ≈ xmax − xmin / 2Mean and spread of repeated readings
Scope of this lesson
- The specific experiments of your PHY 101 laboratory are not listed in the published description; use your laboratory manual for procedures and required uncertainty analysis.
- Least-squares formulas and standard deviations are introduced only by name here.
This is the last lesson in Physics. Back to the Physics contents.
Original study text. Sources and credits.