📚 Testing a Relationship | 检验物理量之间的关系
In CIE A-Level Physics, many experiments ask you to test whether two variables follow a predicted relationship, such as direct proportion, inverse proportion, or a power law. This skill is central to Papers 3 and 5, where you must collect data, plot graphs, and decide whether the evidence supports a given equation.
在 CIE A-Level 物理中，许多实验要求你检验两个变量是否遵循预期的关系，例如正比、反比或幂律关系。这项技能是 Paper 3 和 Paper 5 的核心，你需要采集数据、绘制图形，并判断证据是否支持给定的方程。
1. What Does ‘Testing a Relationship’ Mean? | “检验关系”是什么意思？
When a question asks you to test a relationship, you cannot just take one measurement. You must collect a range of paired values for two variables, plot them, and compare the shape of the graph with the predicted equation. This is a core skill in CIE A-Level Physics Papers 3 and 5.
当题目要求你检验一个关系时，你不能只取一次测量值。你必须为两个变量采集一系列成对数据，绘制图形，并将图形形状与预测方程进行比较。这是 CIE A-Level 物理 Paper 3 和 Paper 5 的核心技能。
A relationship is usually tested by rearranging the equation into a straight-line form y = mx + c. The gradient m and intercept c then tell you the physical constants.
检验关系通常是将方程改写为直线形式 y = mx + c。斜率 m 和截距 c 随后会给出物理常数。
y = mx + c
2. Start from the Expected Equation | 从预期方程出发
Always begin with the physics model you are testing. For example, if you are testing Newton’s second law F = ma, you might keep mass constant and vary force, measuring acceleration. Rearrange to a = F/m, which has the form y = (1/m)x.
始终从你要检验的物理模型开始。例如，如果你正在检验牛顿第二定律 F = ma，你可以保持质量不变，改变力并测量加速度。将其改写为 a = F/m，其形式为 y = (1/m)x。
This step identifies what should be plotted on each axis and what the gradient should represent. Without this step, you may plot the wrong variables and make the relationship impossible to interpret.
这一步确定了每个坐标轴应绘制什么，以及斜率应代表什么。没有这一步，你可能会绘制错误的变量，使关系无法解释。
3. Direct Proportionality: y = kx | 正比关系：y = kx
If the theory predicts y ∝ x, the equation is y = kx. A graph of y against x should be a straight line passing through the origin. If the line is straight but does not pass through the origin, there may be a systematic error or an extra constant in the model.
如果理论预测 y ∝ x，则方程为 y = kx。y 对 x 的图形应是一条经过原点的直线。如果直线是直的但不经过原点，则可能存在系统误差或模型中多了一个常数项。
The gradient of the line is the constant k. For example, charge Q against potential difference V for a capacitor gives Q = CV, so the gradient is capacitance C.
直线的斜率就是常数 k。例如，电容器的电荷量 Q 对电势差 V 的图给出 Q = CV，因此斜率就是电容 C。
y = kx
4. Linear Relationships with an Intercept | 带截距的线性关系
Many A-Level equations have a constant term, such as v = u + at, E = V + Ir, or s = ut + ½at² (when treated appropriately). In these cases, plot the dependent variable on the y-axis and the independent variable on the x-axis. A straight line is expected, but it should not be forced through the origin.
许多 A-Level 方程含有常数项，例如 v = u + at、E = V + Ir 或 s = ut + ½at²（在适当处理时）。在这些情况下，将因变量绘在 y 轴，自变量绘在 x 轴。预期得到一条直线，但不应强行通过原点。
The y-intercept gives the constant term, and the gradient gives the coefficient of the variable. For E = V + Ir, plotting terminal p.d. E against current I gives a straight line with gradient −r and intercept V (where signs depend on convention).
y 截距给出常数项，斜率给出变量的系数。对于 E = V + Ir，将路端电压 E 对电流 I 作图，得到斜率为 −r、截距为 V 的直线（符号取决于约定）。
v = u + at
5. Inverse Proportion: y = k/x | 反比关系：y = k/x
If the model predicts y ∝ 1/x, a direct plot of y against x gives a rectangular hyperbola. This shape is difficult to analyse accurately, so you should test the relationship by plotting y against 1/x.
如果模型预测 y ∝ 1/x，直接绘制 y 对 x 的图会得到一条双曲线。这种形状难以精确分析，因此你应通过绘制 y 对 1/x 的图来检验关系。
If y against 1/x is a straight line through the origin, inverse proportion is confirmed. For example, pressure P against volume V for a fixed mass of gas at constant temperature gives P = k/V, so plotting P against 1/V gives a straight line.
如果 y 对 1/x 的图是一条经过原点的直线，则反比关系得到证实。例如，一定质量气体在恒温下的压强 P 对体积 V 给出 P = k/V，因此绘制 P 对 1/V 的图会得到一条直线。
y = k / x
6. Power Laws and Logarithmic Plots | 幂律关系与对数图
Some relationships involve a power, such as T = 2π√(L/g) for a simple pendulum. Squaring both sides gives T² = 4π²L/g, which is a straight line if T² is plotted against L.
有些关系涉及幂，例如单摆的 T = 2π√(L/g)。将两边平方得到 T² = 4π²L/g，如果绘制 T² 对 L 的图，它是一条直线。
If the power is unknown, write the relationship as y = kxⁿ. Take logarithms to obtain log y = n log x + log k. A graph of log y against log x is a straight line with gradient n and intercept log k.
如果幂未知，可将关系写成 y = kxⁿ。取对数得到 log y = n log x + log k。log y 对 log x 的图是一条直线，斜率为 n，截距为 log k。
log y = n log x + log k
| Relationship | Equation | Plot | Gradient |
|---|---|---|---|
| Direct proportion | y = kx | y vs x | k |
| Inverse proportion | y = k/x | y vs 1/x | k |
| Power law | y = kxⁿ | log y vs log x | n |
| Pendulum | T² = 4π²L/g | T² vs L | 4π²/g |
| Ohm’s law | V = IR | V vs I | R |
7. Experimental Design and Variable Control | 实验设计与变量控制
Choose a sensible range for the independent variable. The range should be wide enough to reveal the shape of the relationship, but not so wide that other effects become significant. For example, when testing Hooke’s law, do not extend the spring beyond its elastic limit.
为自变量选择一个合理的范围。范围应足够宽以揭示关系的形状，但又不能宽到使其他效应变得显著。例如，在检验胡克定律时，不要将弹簧拉伸到超过其弹性极限。
Keep all other variables constant. Record their values and state how you controlled them. Use repeats and calculate mean values to reduce random error.
保持所有其他变量不变。记录它们的数值，并说明你是如何控制它们的。进行重复测量并计算平均值以减小随机误差。
8. Drawing the Best-Fit Line | 绘制最佳拟合线
Plot your data points accurately, using sharp pencil marks and labelled axes with units. If uncertainty bars are required, draw them appropriately. Then draw a single best-fit straight line or smooth curve through the points.
精确绘制数据点，使用清晰的铅笔标记，并标注坐标轴及单位。如果需要误差棒，请适当绘制。然后在数据点之间绘制一条最佳拟合直线或光滑曲线。
Do not force the line through the origin unless the theoretical equation has no intercept. The line should have roughly equal numbers of points above and below it.
除非理论方程没有截距，否则不要强行让直线通过原点。拟合线上下方的数据点数量应大致相等。
9. Using Error Bars to Support or Reject a Relationship | 利用误差棒支持或否定关系
If you have error bars, you can test the relationship more rigorously. A proposed straight line should pass through every error bar if the uncertainties have been estimated correctly. If it cannot, the relationship may be wrong, or the uncertainties may be underestimated.
如果你有误差棒，可以更严格地检验关系。如果误差估计正确，拟议的直线应通过每一个误差棒。如果不能通过，则关系可能有误，或者误差被低估。
For a relationship that predicts a straight line through the origin, check whether the best-fit line passes through the origin within the uncertainty of the intercept. If the intercept is significantly different from zero, there may be a systematic error.
对于预测过原点的直线关系，检查最佳拟合线是否在截距的不确定度范围内通过原点。如果截距显著不为零，则可能存在系统误差。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
One common mistake is using too small a range of data, which makes it impossible to distinguish between different relationships. Another is plotting the variables the wrong way round, so the gradient no longer equals the intended constant.
一个常见错误是使用的数据范围太小，导致无法区分不同的关系。另一个常见错误是将变量绘制反了，使斜率不再等于预期的常数。
Always convert units to SI before plotting. State the gradient with correct units, and use the gradient to calculate the physical constant by comparing it with the theoretical expression.
绘图前务必将单位转换为国际单位制。用正确的单位表示斜率，并通过将斜率与理论表达式进行比较来计算物理常数。
When a question asks ‘test the relationship’, you should write a conclusion in terms of the data: whether the graph is a straight line, whether it passes through the origin, and what the gradient represents.
当题目要求“检验关系”时，你应根据数据写出结论：图形是否为直线，是否通过原点，以及斜率代表什么。
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