Read the core explanations from SHM Practical Investigation before testing yourself from memory.
1. Why This Practical Matters
Picture a pendulum clock ticking steadily on a wall, or a car suspension absorbing a bump. Both are governed by the same physics: simple harmonic motion. Required Practical 7 asks you to measure that motion directly — and the data you collect can verify some of the most elegant equations in A-level physics.
The practical has two distinct systems: a mass on a spring and a simple pendulum. Each one tests a different relationship, and examiners expect you to know both setups inside out — variables, method, analysis, and evaluation. This lesson walks through every layer of that.
2. The Physics Behind the Practical
Before touching any apparatus, it helps to know what you are actually measuring and why the equations take the form they do. For a mass-spring system, the restoring force obeys Hooke's law: $F = -kx$, where $k$ is the spring constant and $x$ is the displacement. Combining this with Newton's second law gives acceleration proportional to displacement — the defining condition for SHM.
For a simple pendulum, the restoring force is the component of gravity acting along the arc. For small angles (below about 10°), $\sin\theta \approx \theta$ in radians, which again gives acceleration proportional to displacement. This small-angle approximation is a critical assumption — violate it and the motion is no longer truly SHM.
Notice that the period of a mass-spring system depends on mass and spring constant only — not on the amplitude of oscillation. This is a key feature of SHM and something examiners test directly.
For the pendulum, the period depends on length and gravitational field strength only — not on the mass of the bob and not on the amplitude (provided the angle stays small). This independence from mass surprises many students and is a favourite examiner target.
3. Required Practical 7 — Mass-Spring System
The aim of the mass-spring investigation is to find how the period of oscillation varies with mass, and to use that relationship to determine the spring constant $k$. Setting up the apparatus carefully is the foundation of reliable data.
The choice to plot $T^2$ against $m$ rather than $T$ against $m$ is deliberate. Plotting $T$ against $m$ would give a curve, which is much harder to analyse. Linearising the relationship by squaring both sides of the period equation turns a curve into a straight line — and a straight line gives you a gradient you can use to calculate $k$ directly.
Worked example: Determining Spring Constant from a T² vs m Graph
A student plots $T^2$ against $m$ for a mass-spring system and obtains a straight line with gradient $0.40$ s² kg$^{-1}$. Calculate the spring constant $k$.
- $T^2 = \frac{4\pi^2}{k} \cdot m$ — This is obtained by squaring both sides of $T = 2\pi\sqrt{m/k}$. The gradient of $T^2$ vs $m$ equals $\frac{4\pi^2}{k}$. [M1]
- $\text{gradient} = \frac{4\pi^2}{k} = 0.40 \text{ s}^2 \text{ kg}^{-1}$ — Read the gradient directly from the graph. [M1]
- $k = \frac{4\pi^2}{0.40}$ — Divide $4\pi^2$ by the gradient value.
- $4\pi^2 = 4 \times 9.8696... \approx 39.48$ — $\pi^2 \approx 9.870$, so $4 \times 9.870 = 39.48$.
- $k = \frac{39.48}{0.40} = 98.7 \text{ N m}^{-1}$ — $39.48 \div 0.40 = 98.7$. Units: $\frac{\text{(dimensionless)}}{\text{s}^2 \text{ kg}^{-1}} = \text{kg s}^{-2} = \text{N m}^{-1}$. [A1]
Answer: $k \approx 98.7 \text{ N m}^{-1}$ (to 3 significant figures)
4. Required Practical 7 — Simple Pendulum
The pendulum investigation follows the same logic as the mass-spring experiment: vary one quantity, measure the period, linearise the relationship, and extract a physical constant from the gradient. Here, the variable is the length of the pendulum, and the constant you extract is $g$, the gravitational field strength.
Measuring the length $L$ accurately is the single biggest source of systematic error in this experiment. The length must be measured from the pivot point to the centre of mass of the bob — not to the top or bottom of the bob. Using a vernier calliper to find the bob's diameter and adding the radius to the string length is the correct technique.
Worked example: Determining g from a T² vs L Graph
A student's $T^2$ vs $L$ graph has a gradient of $4.02$ s² m$^{-1}$. Calculate the value of $g$ obtained from this experiment.
- $T^2 = \frac{4\pi^2}{g} \cdot L$ — Squaring $T = 2\pi\sqrt{L/g}$ gives this linear form. Gradient $= \frac{4\pi^2}{g}$. [M1]
- $\frac{4\pi^2}{g} = 4.02 \text{ s}^2 \text{ m}^{-1}$ — The gradient is read from the best-fit straight line on the graph. [M1]
- $g = \frac{4\pi^2}{4.02}$ — Divide $4\pi^2$ by the gradient.
- $4\pi^2 = 4 \times 9.8696 \approx 39.48$ — $\pi^2 \approx 9.870$; $4 \times 9.870 = 39.48$.
- $g = \frac{39.48}{4.02} \approx 9.82 \text{ m s}^{-2}$ — $39.48 \div 4.02 = 9.82$. This is close to the accepted value of $9.81$ m s$^{-2}$, suggesting the experiment was well-controlled. [A1]
Answer: $g \approx 9.82 \text{ m s}^{-2}$ (to 3 significant figures)
5. Graphical Analysis and Linearisation
Both investigations rely on the same analytical strategy: linearise the period equation so that a graph of $T^2$ against the variable quantity produces a straight line through the origin. The gradient of that line contains the physical constant you want to determine.
When drawing the best-fit line, use a ruler and aim for the line that minimises the total distance of all points from the line — not a line that passes through every point. Anomalous points should be identified, circled, and excluded from the best-fit line, but they must still be shown on the graph.
To find the gradient, draw a large triangle on the best-fit line — as large as possible to reduce the percentage uncertainty in the gradient calculation. Read off the coordinates of the two triangle vertices from the axes, and calculate $\text{gradient} = \frac{\Delta(T^2)}{\Delta L}$ or $\frac{\Delta(T^2)}{\Delta m}$ as appropriate. Always include units in the gradient.
6. Uncertainty, Reliability, and Evaluation
Evaluating your method is worth significant marks in AQA questions. You need to identify specific sources of uncertainty, explain how they affect the result, and suggest concrete improvements — not vague ones like 'be more careful'.
| Source of uncertainty | System affected | Effect on result | Improvement |
|---|
| Reaction time when starting/stopping stopwatch (~0.2 s) | Both | Increases absolute uncertainty in total time; reduced by timing more oscillations | Use a light gate and data logger to remove human reaction time |
| Measuring L to the centre of the bob | Pendulum | Systematic error if radius of bob is not added; underestimates L | Measure bob diameter with vernier calliper; add radius to string length |
| Amplitude exceeding 10° | Pendulum | Small-angle approximation breaks down; period increases slightly above predicted value | Use a protractor or measure horizontal displacement to keep angle below 10° |
| Spring oscillating non-vertically (swinging as well as bouncing) | Mass-spring | Period measured is not purely for vertical SHM; introduces scatter | Displace mass carefully along the vertical axis only; use a guide rod if available |
| Mass of the spring | Mass-spring | Effective oscillating mass is greater than the attached mass; $T^2$ vs $m$ line does not pass through origin | Account for spring mass by extrapolating the line to find the effective additional mass |
Reliability is improved by repeating each timing at least three times and calculating a mean period. Validity is ensured by keeping all control variables constant — particularly the amplitude, which must remain small throughout both investigations.
7. Comparing the Two Systems
Both systems demonstrate SHM, but they test different physical relationships. Understanding the comparison between them is useful for evaluation questions and for checking your conceptual understanding of the period equations.
| Feature | Mass-spring system | Simple pendulum |
|---|
| Period equation | $T = 2\pi\sqrt{m/k}$ | $T = 2\pi\sqrt{L/g}$ |
| Variable investigated | Mass $m$ (spring constant $k$ fixed) | Length $L$ (g fixed) |
| Physical constant extracted | Spring constant $k$ | Gravitational field strength $g$ |
| Graph plotted | $T^2$ vs $m$ | $T^2$ vs $L$ |
| Gradient equals | $4\pi^2/k$ | $4\pi^2/g$ |
| Key assumption | Spring obeys Hooke's law; spring mass negligible | Small-angle approximation ($\theta < 10°$); string is inextensible and massless |
| Effect of increasing the variable | Period increases (heavier mass oscillates more slowly) | Period increases (longer pendulum oscillates more slowly) |
| Effect of changing amplitude | No effect on period (SHM property) | No effect on period provided angle stays below ~10° |
8. Exam Technique and Mark-Winning Guidance
AQA questions on Required Practical 7 follow predictable patterns. They ask you to describe the method, identify variables, sketch or interpret a graph, calculate a physical constant from a gradient, or evaluate the experiment. Each of these has a specific structure that earns marks.
When describing the method, be specific about numbers: state that you time 20 oscillations (not 'several'), that the amplitude is kept below 10° (not 'small'), and that you repeat each measurement at least three times to find a mean. Vague method descriptions lose marks.