Physics Practical Skills: Graphs, Slope, Errors and Precautions

Marks You Can Earn in Every Practical

In WAEC and NECO physics practical papers, and in university laboratory reports, a large share of the marks does not depend on which experiment you are given. Examiners reward the same skills every time: recording readings correctly in a table, plotting a good graph, finding the slope accurately, using it to calculate a physical quantity, and stating sensible precautions. These skills are easy to learn and easy to lose marks on. This tutorial covers physics practical skills step by step: tables of readings, significant figures and units, choosing scales and plotting graphs, drawing the line of best fit, calculating the slope and intercept, deducing physical quantities from straight-line graphs, errors and uncertainties, and writing precautions, illustrated with a complete simple-pendulum experiment.

The Example Experiment: the Simple Pendulum

A student measures the time t for 20 complete oscillations of a simple pendulum for six different lengths l. The aim is to plot a graph of T² against l and use its slope to find g, the acceleration due to gravity.

The theory is T = 2π√(l/g). Squaring both sides gives:

T² = (4π² ÷ g) × l

This has the form y = mx, a straight line through the origin, with y = T², x = l and slope m = 4π² ÷ g. So g = 4π² ÷ slope.

Step 1: Recording the Readings

In any physics practical, readings should be set out in a table with clear column headings that show each quantity and its unit, such as "l / m" or "t / s". Here are the student's readings, one line per row, in the order l / m, t / s, T / s and T² / s²:

0.400, 25.4, 1.270, 1.61
0.500, 28.4, 1.420, 2.02
0.600, 31.1, 1.555, 2.42
0.700, 33.6, 1.680, 2.82
0.800, 35.9, 1.795, 3.22
0.900, 38.1, 1.905, 3.63

The period is T = t ÷ 20, and the last column is T squared.

Good habits for tables

Take at least six readings, spread evenly over as wide a range as possible.
Record raw readings to the precision of the instrument. A metre rule reads to 0.001 m (1 mm), and a stopwatch to 0.1 s for hand timing.
Keep the number of decimal places consistent down each column of raw readings.
Give calculated values to a sensible number of significant figures, generally the same as, or one more than, the least precise raw reading used.
Repeat readings where possible and record the mean. Timing many oscillations, here 20, reduces the effect of your reaction time on each period.

Step 2: Plotting the Graph

"Plot a graph of T² against l" means that T² goes on the vertical (y) axis and l on the horizontal (x) axis: the quantity before "against" is always vertical.

Label both axes with the quantity and unit, such as "T² / s²" and "l / m".
Choose sensible scales so that the points fill at least half the graph paper in each direction. Use easy scales such as 1, 2 or 5 units per large square (or multiples of ten of these). Avoid awkward scales such as 3 or 7 units per square, which make plotting and reading off values slow and error-prone.
The origin need not be included unless you need the intercept, or the question asks for it. Here, including it lets you check that the line passes through it.
Plot points accurately as small dots with circles round them, or as small neat crosses, to within half a small square.
Give the graph a title, such as "Graph of T² against l".

Step 3: Drawing the Line of Best Fit

Real readings contain small random errors, so the points will not lie exactly on a line. Draw a single thin straight line of best fit with a ruler:

Roughly equal numbers of points should lie on each side of the line, and the points should be scattered evenly along it.
Do not join the dots point to point, and do not force the line through the first and last points or through the origin unless the data support it.
A point that lies far from the line is an anomalous reading. Ignore it when drawing the line, and if possible repeat that measurement.

Step 4: Calculating the Slope

The slope (gradient) is the rise divided by the run:

slope = Δy ÷ Δx = (y₂ − y₁) ÷ (x₂ − x₁)

Draw a large right-angled triangle on the line, covering more than half of it, because a large triangle reduces the error in reading the values.
Take the two points from the line, not from the table of readings, unless those readings lie exactly on the line.
Show the triangle on the graph, and write down the coordinates you read.
Give the slope its unit: the y unit divided by the x unit.

Worked example 1

The best-fit line for the pendulum data passes very close to (0.400, 1.61) and (0.900, 3.63). Find the slope and hence g.

Slope = (3.63 − 1.61) ÷ (0.900 − 0.400) = 2.02 ÷ 0.500 = 4.04 s²/m.

g = 4π² ÷ 4.04 = 39.48 ÷ 4.04 ≈ 9.77 m/s².

A least-squares calculation over all six points gives a slope of 4.03 s²/m and g ≈ 9.80 m/s², close to the accepted value of 9.81 m/s². The graph method uses all the readings at once, which averages out random errors, and it is more reliable than calculating g from any single reading.

The intercept

The intercept is where the line crosses the y-axis (at x = 0). For a straight line y = mx + c, it is c. Here, the theory predicts that the line passes through the origin, and the best-fit intercept is about 0.001 s², which is effectively zero. A significant non-zero intercept would suggest a systematic error, such as measuring the length to the top of the bob instead of to its centre.

Step 5: Linearising Other Relationships

Straight lines are easy to test and measure, so physicists rearrange formulas into the form y = mx + c before plotting. Some common examples:

The pendulum: plot T² against l. The slope is 4π²/g.
Ohm's law: plot V against I. The slope is the resistance R.
A cell's internal resistance: V = E − Ir, so plot V against I. The intercept is E, and the slope is −r.
Hooke's law: plot the force F against the extension e. The slope is the spring constant k.
The lens formula: 1/v + 1/u = 1/f, so plot 1/v against 1/u. Both intercepts equal 1/f.
Refraction: plot sin i against sin r. The slope is the refractive index n.
Radioactive decay: N = N₀e^(−λt), so ln N = ln N₀ − λt. Plot ln N against t. The slope is −λ.

Errors and Uncertainties

Types of error

Random errors scatter readings unpredictably on either side of the true value, for example from reaction time when starting and stopping a stopwatch, or from judging a reading between scale marks. They are reduced by repeating readings and averaging, and by plotting graphs.
Systematic errors shift every reading in the same direction, for example from a zero error on a micrometer, a stopwatch that runs fast, or always reading a scale from the same wrong angle. Repeating does not reduce them. They must be found and corrected, and they often show up as an unexpected intercept on a graph.
Parallax error occurs when a scale is read from an angle instead of straight on. Keep your eye level with, and perpendicular to, the reading.

Percentage uncertainty

Percentage uncertainty = (uncertainty ÷ measured value) × 100%

When quantities are multiplied or divided, add their percentage uncertainties. When a quantity is raised to a power, multiply its percentage uncertainty by the power.

Worked example 2

A student times 20 oscillations as 25.4 s, with an uncertainty of ±0.2 s from reaction time, using a length of 40.0 cm ± 0.1 cm. Estimate the percentage uncertainty in g calculated from g = 4π²l/T².

Uncertainty in t (and so in T): 0.2 ÷ 25.4 × 100 ≈ 0.79%. T is squared, so this contributes 2 × 0.79 ≈ 1.6%.

Uncertainty in l: 0.1 ÷ 40.0 × 100 = 0.25%.

Total uncertainty in g ≈ 1.6 + 0.25 ≈ 1.8%. Timing is the main source of error, which is why timing more oscillations helps: 50 oscillations instead of 20 would reduce it to well under 1%.

Writing Precautions

A precaution is a specific action taken to improve the accuracy of the results. WAEC rewards precautions stated in the past tense, specific to the experiment and phrased as actions. Vague statements such as "I was careful" or "I avoided errors" earn no marks.

Precautions for the pendulum experiment

I ensured the oscillations were of small amplitude (less than about 10°).
I ensured the pendulum swung in a vertical plane, without spinning or moving in an ellipse.
I started timing as the bob passed through its lowest point, and counted the oscillations carefully.
I measured the length from the point of suspension to the centre of the bob.
I avoided parallax error when reading the metre rule, and repeated the timings and took the mean.

General precautions for other experiments

Electricity: I ensured clean, tight connections. I opened the key when not taking readings, to prevent the wire heating up. I checked the meters for zero error.
Optics: I ensured the object, lens and screen were in line. I obtained a sharp image before measuring. I used pins as far apart as possible, to trace rays accurately.
Heat: I stirred the liquid before reading the temperature. I lagged the calorimeter to reduce heat loss. I read the thermometer with my eye level with the mercury meniscus.
Mechanics: I ensured the metre rule balanced horizontally. I made sure the spring was not stretched beyond its elastic limit. I tapped the apparatus gently to reduce friction before reading.

Common Mistakes in Physics Practicals

Missing units in table headings, on axes, or on the slope.
Awkward graph scales, or points squeezed into a small corner of the paper.
Joining the points dot to dot instead of drawing a single line of best fit.
Using a small triangle, or readings from the table that are not on the line, to calculate the slope.
Vague precautions, such as "the experiment was done carefully".
Too many or too few significant figures, such as giving g = 9.774193 m/s² from readings taken to three significant figures.

Physics Practical Skills: Practice Questions

A graph of V against I for a cell is a straight line with an intercept of 1.5 V and a slope of −0.4 V/A. Find the e.m.f. and internal resistance of the cell.
A graph of the extension of a spring (in metres) against the load (in newtons) has a slope of 0.025 m/N. Find the spring constant.
Explain why a student should time 20 oscillations instead of just one.
A length of 25.0 cm is measured with an uncertainty of ±0.1 cm. Find the percentage uncertainty.
Distinguish between a random error and a systematic error, with one example of each.
State two precautions for an experiment to determine the refractive index of a glass block using pins.

Answers: 1. E = 1.5 V, and r = 0.4 Ω. 2. k = 1 ÷ 0.025 = 40 N/m (this graph has extension on the vertical axis, so the slope is 1/k). 3. Timing many oscillations reduces the percentage error caused by reaction time when starting and stopping the stopwatch. 4. 0.1 ÷ 25.0 × 100 = 0.4%. 5. A random error scatters readings both ways, such as reaction time. A systematic error shifts every reading the same way, such as a zero error on a meter. 6. Any two of: I placed the pins vertically and well apart; I viewed the bases of the pins to avoid parallax; I used a sharp pencil to trace the outline of the block; I kept the block in the same position throughout.

Conclusion

Good practical work follows the same steps every time. Record readings in a table with quantities and units, at the instrument's precision. Plot the dependent variable against the independent one on well-chosen scales, and draw a single line of best fit. Find the slope from a large triangle on the line, with its unit, and use the linearised formula, such as T² = (4π²/g)l, to deduce the physical quantity. Understand random and systematic errors and estimate percentage uncertainties, then state specific, action-based precautions. These skills earn marks in every physics practical examination.