📚 Physics Experimental Investigation | 物理实验探究
Physics is fundamentally an experimental science. No matter how elegant a theoretical model appears, it must be tested and validated through carefully designed investigations. An experimental inquiry involves identifying a physical relationship, controlling relevant variables, collecting quantitative data, analysing uncertainties, and drawing evidence-based conclusions. Mastering these skills not only prepares you for practical examinations but also develops the scientific mindset required at university and beyond.
物理学本质上是一门实验科学。无论一个理论模型多么优美，都必须通过精心设计的实验来检验和验证。一项实验探究包括确定物理关系、控制相关变量、收集定量数据、分析不确定度并得出基于证据的结论。掌握这些技能不仅能为实验考试做好准备，也能培养在大学及更高阶段所需的科学思维。
1. The Nature of Experimental Physics | 实验物理学的本质
Experimental physics is not simply about following a set of instructions; it is a process of discovery. It begins with a question or a hypothesis derived from an observed phenomenon. The investigator then designs a procedure to test that hypothesis, ensuring that the results are reproducible and reliable. A successful experiment reveals a quantitative relationship between physical quantities, often expressed as a mathematical law.
实验物理并不仅仅是按部就班地操作，而是一个发现的过程。它始于从观察到的现象中提出的问题或假设。研究者随后设计一套程序来检验该假设，并确保结果具有可重复性和可靠性。一个成功的实验能揭示物理量之间的定量关系，通常以数学定律的形式表达。
Central to any investigation is the concept of measurement. Every measurement has an associated uncertainty, and the quality of an experiment is judged not by the absence of error but by how well those uncertainties are understood and minimised. This is what distinguishes a rough estimate from a precise scientific measurement.
任何探究的核心都是测量。每一次测量都伴随着不确定度，而评判实验质量的标准并非完全没有误差，而是对这些不确定度的理解与最小化程度。这正是粗略估计与精确科学测量之间的区别所在。
2. Planning and Designing an Experiment | 实验的计划与设计
A well-planned experiment starts with a clear statement of the aim. You must specify the independent variable (the quantity you will change), the dependent variable (the quantity you will measure), and the control variables (all other factors that must be kept constant to ensure a fair test). A preliminary trial often helps to determine a suitable range and interval for the independent variable.
一个精心计划的实验始于清晰的目标陈述。你必须明确自变量（你将改变的量）、因变量（你将测量的量）以及控制变量（为确保公平测试而必须保持恒定的所有其他因素）。初步试验通常有助于确定自变量的合适范围和间隔。
Choosing appropriate apparatus is equally important. For example, if you need to measure the diameter of a wire, a micrometer screw gauge offers far greater precision than a metre ruler. You should also plan how many repeats to take. At least three readings for each value of the independent variable are recommended, with more if time permits, to reduce the impact of random errors and allow the calculation of a mean.
选择合适的仪器同样重要。例如，如果需要测量一根金属丝的直径，千分尺比米尺能提供高得多的精度。你还应该计划好要重复测量多少次。建议对自变量的每个取值至少读取三次数据，若时间允许则更多，以减少随机误差的影响并能够计算平均值。
3. Variables and Controls | 变量与控制
The independent variable is the one deliberately manipulated by the experimenter. The dependent variable responds to this change and is measured. All other variables that could influence the dependent variable are called control variables. Keeping these constant is essential for isolating the relationship under investigation. For instance, when investigating the period of a pendulum, the length is independent, the period is dependent, while the mass of the bob and the amplitude of swing (if kept small) are controls.
自变量是实验者有意操纵的量。因变量则对此变化作出反应并被测量。所有其他可能影响因变量的量称为控制变量。保持这些变量恒定对于分离出所研究的特定关系至关重要。例如，在研究单摆周期时，摆长是自变量，周期是因变量，而摆球质量和摆动幅度（如果保持较小）则是控制变量。
Sometimes variables cannot be easily controlled, such as air currents or temperature fluctuations in a room. In such cases, you should monitor them and note them as potential sources of error. A thorough evaluation always discusses how well each control variable was managed.
有时某些变量难以控制，比如室内的气流或温度波动。在这种情况下，你应该监控它们并作为潜在的误差来源记录下来。全面的评估总是会讨论每个控制变量的管理情况。
4. Data Collection Techniques | 数据收集技巧
Data should be recorded in a table immediately, with clear headings that include the quantity and its unit. Using a ruler to draw a neat table on plain paper is perfectly acceptable in an examination. For each measurement, record the raw data; never try to ‘correct’ a reading that appears out of line with the others unless you have a justifiable instrumental reason. Such outliers should be noted and repeated if possible.
数据应立即记录在表格中，表头应明确注明物理量及其单位。在考试中，用尺子在白纸上画出整齐的表格是完全可行的。对于每次测量，都要记录原始数据；除非有合理的仪器故障原因，否则绝不要试图“修正”一个看似与其他数据不符的读数。这类异常值应予以注意，并在可能的情况下重复测量。
When measuring time intervals, it is usually better to measure multiple oscillations (e.g., 10 or 20 periods) and then divide, rather than timing a single period. This technique reduces the relative uncertainty contributed by human reaction time. Using a stopwatch with a resolution of 0.01 s still carries a reaction time uncertainty of about 0.2 s, so timing 20 swings makes that uncertainty a much smaller fraction of the total recorded time.
在测量时间间隔时，通常更可取的是测量多次振荡（例如 10 或 20 个周期）然后相除，而不是只测一个周期。这种技巧可以降低由人体反应时间带来的相对不确定度。即使使用分辨率为 0.01 s 的秒表，反应时间的不确定度仍约为 0.2 s，因此如果计时 20 次摆动，这一不确定度在总记录时间中所占的比例就会小很多。
5. Uncertainty and Error Analysis | 不确定度与误差分析
No measurement is exact. The absolute uncertainty of a single reading is usually taken as half the smallest division of the measuring instrument, unless the manufacturer specifies otherwise. For a digital instrument, it is the last significant digit. When a mean value is calculated from several repeats, the absolute uncertainty can be estimated as half the range of the values, i.e., (maximum – minimum)/2.
没有哪个测量是绝对精确的。单次读数的绝对不确定度通常取为测量仪器最小分度的一半，除非制造商另有说明。对于数字式仪器，则是最后一位有效数字。当由多次重复测量计算出平均值时，绝对不确定度可以估算为数值范围的一半，即 (最大值 – 最小值)/2。
The percentage uncertainty gives a clearer idea of precision: (absolute uncertainty / measured value) × 100%. If your final result depends on several measured quantities, you must combine uncertainties. For addition or subtraction, absolute uncertainties add. For multiplication or division, percentage uncertainties add. For a power relationship like y = k xⁿ, the percentage uncertainty in y is n times the percentage uncertainty in x.
百分比不确定度能更清晰地反映精度：(绝对不确定度 / 测量值) × 100%。如果最终结果依赖于多个被测量，你就必须合成不确定度。对于加法或减法，绝对不确定度相加。对于乘法或除法，百分比不确定度相加。对于像 y = k xⁿ 这样的幂函数关系，y 的百分比不确定度是 x 的百分比不确定度的 n 倍。
Δg / g = ΔL / L + 2(ΔT / T)
The above formula shows the propagation of uncertainties when determining g from a pendulum experiment, where g = 4π²L / T². Since T is squared, its percentage uncertainty is doubled.
上述公式展示了从单摆实验中确定 g 时的不确定度传播，其中 g = 4π²L / T²。由于 T 被平方，它的百分比不确定度要加倍。
6. Graphical Analysis | 图表分析
Plotting a graph is one of the most powerful ways to reveal a relationship. The independent variable goes on the horizontal axis and the dependent on the vertical axis. Both axes must be labelled with the quantity and unit, and scales should be linear, convenient, and occupy more than half the graph paper. Data points should be plotted with small, neat crosses and should not be joined dot-to-dot unless instructed otherwise.
绘制图表是揭示关系的最有效方法之一。自变量放在横轴，因变量放在纵轴。两条轴都必须标注物理量和单位，并且标度应该线性、方便且占据图表纸一半以上的面积。数据点应用细小的十字精确标出，除非另有指示，否则不应逐点连线。
A best-fit line should be drawn to reflect the trend. This line may be straight or curved, and there should be an even spread of points on both sides. If the relationship is expected to be linear, the gradient and y-intercept often carry physical meaning. For example, a graph of T² against L for a pendulum should pass through the origin, and its gradient equals 4π²/g. The gradient is calculated using a large triangle, not data points, to minimise reading errors.
应绘制一条最佳拟合线以反映趋势。这条线可以是直线或曲线，且两侧分布的点应大致均匀。如果预期关系为线性，那么斜率和 y 轴截距通常具有物理意义。例如，单摆实验中 T² 对 L 的图线应通过原点，其斜率等于 4π²/g。计算斜率时应使用一个较大的三角形而非原始数据点，以减小读数误差。
7. Evaluating Results and Drawing Conclusions | 评估结果与得出结论
Once the graph is plotted and the gradient calculated, you should use the gradient to determine the desired quantity—for instance, g from the pendulum graph. Your final answer must be quoted with its absolute uncertainty, derived from your uncertainty propagation calculations. Compare your value with the accepted literature value, and calculate the percentage difference. A difference within your experimental uncertainty suggests your result is consistent with the accepted value.
绘制完图表并计算出斜率后，你应当利用斜率来确定所求的量——例如从单摆图线中求出 g。你的最终答案必须附上根据不确定度传播计算得出的绝对不确定度。将你的值与公认的文献值进行比较，并计算百分比差异。如果差异落在你的实验不确定度范围之内，就表明你的结果与公认值一致。
A fair conclusion does not claim perfection but states what the experiment demonstrates. It may read: ‘The straight-line graph confirms that T² is proportional to L, and the calculated value of g is 9.7 ± 0.4 m s⁻², which agrees within experimental uncertainty with the accepted value of 9.81 m s⁻².’ Any discrepancy should be discussed in terms of systematic or random errors.
公正的结论不会宣称完美，而是陈述实验所证明的内容。它可以这样表述：“该直线图线证实了 T² 与 L 成正比，计算出的 g 值为 9.7 ± 0.4 m s⁻²，在实验不确定度范围内与公认值 9.81 m s⁻² 一致。”任何偏差都应根据系统误差或随机误差加以讨论。
8. Common Pitfalls and How to Avoid Them | 常见误区及避免方法
One of the most frequent mistakes is ignoring zero error. Always check a micrometer or vernier caliper for zero reading before taking measurements. Another common error is measuring the length of a pendulum string from the support to the top of the bob, rather than to the centre of mass of the bob. This introduces a systematic error that shifts all points on the graph.
最常见的错误之一是忽略零误差。在测量之前，务必检查千分尺或游标卡尺的零位读数。另一个常见错误是测量单摆的线长时从悬点到摆球的顶部，而不是到摆球的质心。这会引入系统误差，使图上的所有点发生平移。
Students often forget to record the instrument’s resolution and the uncertainty in measurements. Simply writing down numbers without units or without indicating repeat readings undermines the validity of the data. In graph work, a frequent error is choosing an awkward scale (e.g., each small square = 3 units) or failing to label axes correctly. Always test your scale by seeing if you can read intermediate values easily.
学生常常忘记记录仪器的分辨率和测量中的不确定度。只写下数字而没有单位，或不标明重复读数，都会损害数据的有效性。在图表工作中，一个常见错误是选择了难读的标度（例如每小格 = 3 个单位）或未能正确标记坐标轴。一定要通过检查能否轻松读出中间值来测试你的标度。
9. Safety in the Laboratory | 实验室安全
Safety must never be an afterthought. All investigations should be preceded by a risk assessment. While many physics experiments involve low-risk equipment, hazards can still arise from masses falling, springs under tension, hot surfaces, or electrical circuits. Always keep the workspace clear, tie back long hair, and wear appropriate eye protection when required.
安全绝不应是事后才考虑的事情。所有探究开始前都应进行风险评估。尽管许多物理实验所用设备风险较低，但下落的重物、张紧的弹簧、热表面或电路仍可能带来危险。务必保持工作区域整洁，束好长发，并在必要时佩戴适当的护目镜。
When using electrical equipment, check that leads and connections are insulated and that the voltage does not exceed the rating of components. For experiments involving heavy masses, ensure that the clamp stands and supports are stable and that there is a soft landing area if something slips. A careful scientist anticipates what could go wrong and takes steps to prevent it.
使用电气设备时，要检查导线和连接处是否绝缘，并确保电压不超过元件的额定值。对于涉及重物的实验，要确保铁架台和支架稳固，并在有滑落风险时设置软着陆区域。严谨的科学家会预想可能出现的问题并采取措施加以防范。
10. Sample Investigation: Measuring g Using a Simple Pendulum | 示例探究：用单摆测量 g
In this classic experiment, the independent variable is the length L of a light, inextensible string, and the dependent variable is the period T of the pendulum. A metal bob of fixed mass is used, and the amplitude of swing is kept small (< 10°) to satisfy the simple harmonic motion approximation. The compound pendulum formula is not needed here.
在这个经典实验中，自变量是轻质且不可伸长的细线的长度 L，因变量是单摆的周期 T。使用一个固定质量的金属摆球，并保持摆动幅度很小 (< 10°) 以满足简谐运动近似。这里不需要使用复摆公式。
| Length L / m | t₁ (20 T) / s | t₂ (20 T) / s | Mean T / s | T² / s² |
| 0.600 | 31.14 | 31.20 | 1.558 | 2.427 |
| 0.800 | 35.98 | 36.06 | 1.801 | 3.244 |
| 1.000 | 40.24 | 40.30 | 2.014 | 4.056 |
| 1.200 | 44.08 | 44.12 | 2.205 | 4.862 |
The table above shows specimen data. The uncertainty in L is taken as ±0.001 m (using a metre ruler with a set square to align the bob). The uncertainty in each timing is dominated by human reaction time, estimated at ±0.2 s for each 20-swing measurement. Hence the absolute uncertainty in mean T is about 0.01 s, calculated from half the range divided by 20.
上表展示了示例数据。L 的不确定度取为 ±0.001 m（使用米尺配合直角尺对准摆球）。每次计时的不确定度主要由人体反应时间决定，估计为每次 20 个摆动测量 ±0.2 s。因此，平均周期 T 的绝对不确定度约为 0.01 s，由极差的一半除以 20 计算得出。
A graph of T² (vertical axis) against L (horizontal axis) is plotted. The gradient is found using a large triangle, yielding a value of 4.05 s² m⁻¹. Using the relation T² = (4π²/g) L, the gradient equals 4π²/g. Therefore, g = 4π² / gradient = 4 × (3.142)² / 4.05 ≈ 9.75 m s⁻². The percentage uncertainty in the gradient is estimated from the maximum and minimum gradient lines drawn by considering error bars. Suppose the gradient uncertainty is ±0.08 s² m⁻¹, then the percentage uncertainty in gradient is (0.08/4.05) × 100% ≈ 2.0%, giving Δg ≈ 0.2 m s⁻².
绘制 T²（纵轴）对 L（横轴）的图线。使用一个较大的三角形求出斜率，得到 4.05 s² m⁻¹。利用关系式 T² = (4π²/g) L，斜率等于 4π²/g。因此，g = 4π² / 斜率 = 4 × (3.142)² / 4.05 ≈ 9.75 m s⁻²。通过考虑误差棒绘制的最大和最小斜率线来估算斜率的百分比不确定度。假设斜率不确定度为 ±0.08 s² m⁻¹，那么斜率的百分比不确定度为 (0.08/4.05) × 100% ≈ 2.0%，从而 Δg ≈ 0.2 m s⁻²。
The final result is expressed as g = 9.8 ± 0.2 m s⁻². The accepted value of 9.81 m s⁻² lies well within this range, indicating that the experiment is accurate within its claimed precision. To improve, one could use a light gate pair to eliminate reaction-time error and measure the bob’s centre of mass directly. A longer string would also reduce the relative uncertainty in length.
最终结果表示为 g = 9.8 ± 0.2 m s⁻²。公认值 9.81 m s⁻² 完全落在此范围内，表明该实验在其声称的精度内是准确的。若要改进，可以使用一对光闸来消除反应时间误差，并直接测量摆球的质心位置。更长的细线也能减小长度的相对不确定度。
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