Investigating the Behaviour of a Spring Under Load: A-Level Physics Jun 18 Insert 1 | 探究弹簧负载行为：A-Level物理2018年6月材料1实验解析

📚 Investigating the Behaviour of a Spring Under Load: A-Level Physics Jun 18 Insert 1 | 探究弹簧负载行为：A-Level物理2018年6月材料1实验解析

In the June 2018 A-Level Physics examination, Insert 1 presented a classic experimental investigation into the mechanical properties of a metal spring. Candidates were required to analyse data, interpret a force–extension relationship, and evaluate the reliability of the experiment. This article reconstructs the likely scenario, explains the underlying physics principles, and provides a step-by-step guide to handling such data-based questions in an exam context.

在2018年6月的A-Level物理考试中，材料1（Insert 1）展示了一个关于金属弹簧力学性质的经典实验探究。考生需要分析数据、解释力与伸长的关系，并评估实验的可靠性。本文重建了该实验的可能场景，解释其背后的物理原理，并逐步指导如何在考试中处理这类基于数据的题目。

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1. The Experimental Context | 实验背景

The insert described a student suspending different masses from a vertical spring and measuring the subsequent extension. The primary aim was to verify Hooke’s Law and determine the spring constant, k. In addition, the student was asked to identify the point at which the spring no longer obeys Hooke’s Law, known as the elastic limit.

该材料描述了一名学生在竖直悬挂的弹簧上悬挂不同质量的重物并测量相应的伸长量，主要目的是验证胡克定律并测定弹簧劲度系数 k。此外，还需要学生识别弹簧不再遵守胡克定律的点，即弹性极限。

The data provided in Insert 1 included a table of masses, corresponding forces (weight), original length of the spring, final lengths, and calculated extensions. A typical force–extension graph was to be plotted from the data.

材料1中给出的数据包括一个表格，列出了质量、对应的力（重量）、弹簧原长、最终长度以及计算出的伸长量。通常要求根据这些数据绘制力–伸长图。

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2. Apparatus and Setup | 实验仪器与装置

A typical setup for this investigation includes a helical spring suspended from a rigid clamp, a metre rule placed vertically alongside the spring, a weight hanger with slotted masses, and a pointer attached to the bottom of the spring to read the position accurately.

该实验的典型装置包括：悬挂在刚性夹上的螺旋弹簧、垂直放置在弹簧旁的米尺、带有槽码的吊钩，以及固定在弹簧底端用于准确读数的指针。

It is essential to eliminate parallax error by ensuring the eye is level with the pointer when reading the rule. A set square or a mirror behind the rule can assist in aligning the line of sight.

消除视差至关重要，读数时应确保眼睛与指针在同一水平线上。在米尺后放置一把三角板或镜子有助于对齐视线。

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3. Procedure and Data Collection | 实验步骤与数据采集

First, the unstretched length of the spring, L₀, was measured without any load. Masses were then added incrementally, and the new length L was recorded each time. The extension, x, was calculated as x = L – L₀.

首先，在无负载的情况下测量弹簧的原长 L₀。然后逐步增加质量，每次都记录新长度 L。伸长量 x 计算为 x = L – L₀。

The weight of the hanging mass, F = mg, was taken as the applied force. It is common practice to convert mass in grams to kilograms and multiply by 9.81 m s⁻² to obtain force in newtons.

悬挂物体的重量 F = mg 被视为施加的力。通常的做法是将以克为单位的质量转换为千克，并乘以 9.81 m s⁻² 以获得以牛顿为单位的力。

Insert 1 provided a table with columns: mass m / g, weight F / N, length L / mm, and extension x / mm. Candidates were expected to complete missing values and use them for graphing.

材料1给出了一个包含以下列的表格：质量 m / g、重量 F / N、长度 L / mm 和伸长量 x / mm。考生需要补全缺失数值并用于作图。

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4. Analysing the Insert 1 Data Table | 分析材料1的数据表

The supplied data might look like this:

所提供的数据可能如下所示：

Mass / g	Weight / N	Length / mm	Extension / mm
0	0.00	50.0	0.0
100	0.981	56.5	6.5
200	1.962	63.2	13.2
300	2.943	70.0	20.0
400	3.924	76.7	26.7
500	4.905	83.5	33.5
600	5.886	91.0	41.0

From the table, the extension increases proportionally with weight until the last few readings, where a slight deviation suggests approaching the elastic limit.

从表中可以看出，伸长量随重力成比例增加，直到最后几个读数出现轻微偏离，表明接近弹性极限。

When plotting the force–extension graph, the gradient in the linear region gives the spring constant k. A typical question asks: “Determine k from the graph.”

绘制力–伸长图时，线性区域的斜率即为弹簧劲度系数 k。典型的问题是：“根据图线求出 k。”

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5. Graph Plotting and Determination of k | 绘图与求 k 值

Plot force F on the y‑axis and extension x on the x‑axis. The Hooke’s Law region shows a straight line through the origin, obeying F = kx. The gradient of this straight line is k.

将力 F 标在 y 轴上，伸长量 x 标在 x 轴上。符合胡克定律的区域显示一条通过原点的直线，满足 F = kx。这条直线的斜率即为 k。

To calculate the gradient, select two well-separated points on the linear part: (x₁, F₁) and (x₂, F₂). Then k = (F₂ – F₁) / (x₂ – x₁). For the data above, using (0.0, 0.0) and (33.5 mm, 4.905 N), convert mm to m: 33.5 mm = 0.0335 m. So k = 4.905 N / 0.0335 m ≈ 146 N m⁻¹.

计算斜率时，在线性部分选择两个相距较远的点：(x₁, F₁) 和 (x₂, F₂)。然后 k = (F₂ – F₁) / (x₂ – x₁)。对于上述数据，使用 (0.0, 0.0) 和 (33.5 mm, 4.905 N)，将 mm 转换为 m：33.5 mm = 0.0335 m。因此 k = 4.905 N / 0.0335 m ≈ 146 N m⁻¹。

Always state the unit of k: N m⁻¹. Some students forget unit conversion and obtain k in N mm⁻¹; this must be avoided.

务必注明 k 的单位：N m⁻¹。一些学生忘记单位换算，得到 k 的单位为 N mm⁻¹；这必须避免。

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6. Identifying the Limit of Proportionality | 确定比例极限

The point at which the graph begins to curve marks the limit of proportionality. Beyond this, Hooke’s Law is no longer valid, and the spring starts to deform plastically if the elastic limit is exceeded.

图线开始弯曲的点标志着比例极限。超过该点，胡克定律不再成立，并且如果超过弹性极限，弹簧将开始发生塑性形变。

In the data, the 600 g mass gives an extension of 41.0 mm. If we check proportionality: at 500 g, extension per newton = 33.5 / 4.905 = 6.83 mm N⁻¹. At 600 g, extension per newton = 41.0 / 5.886 = 6.97 mm N⁻¹. The ratio has increased, indicating we are leaving the linear region.

在数据中，600 g 的质量产生的伸长量为 41.0 mm。如果我们检查比例关系：在 500 g 时，每牛顿伸长量 = 33.5 / 4.905 = 6.83 mm N⁻¹。在 600 g 时，每牛顿伸长量 = 41.0 / 5.886 = 6.97 mm N⁻¹。比值增大，表明我们正离开线性区域。

The elastic limit may be slightly beyond 500 g. A safe exam answer would note that the spring obeys Hooke’s Law up to at least 500 g, and beyond that the relationship becomes non-linear.

弹性极限可能稍高于 500 g。考试中稳妥的回答是：弹簧至少在 500 g 以下遵守胡克定律，超过该负载后关系变为非线性。

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7. Uncertainty Analysis | 不确定度分析

The metre rule typically has a precision of ±1 mm. When measuring length, the absolute uncertainty in each reading is ±0.5 mm at each end, giving a total uncertainty of ±1 mm in the extension if measured directly as difference.

米尺的典型精度为 ±1 mm。测量长度时，每次读数的绝对不确定度为 ±0.5 mm（首尾两端），因此如果直接测量差值，伸长量的总不确定度为 ±1 mm。

If a digital balance was used to measure mass, the uncertainty might be ±0.1 g. The uncertainty in force arises from the product of mass and g; since g is assumed to be 9.81 m s⁻² exactly, the percentage uncertainty in force is the same as that in mass.

如果使用数字天平测量质量，不确定度可能是 ±0.1 g。力的不确定度来源于质量与 g 的乘积；由于 g 被视为准确的 9.81 m s⁻²，力的百分比不确定度与质量的百分比不确定度相同。

For the largest force (5.886 N), % uncertainty in mass = 0.1/600 × 100% ≈ 0.017%, which is negligible compared to the length uncertainty. Therefore, the dominant source of error is the measurement of extension.

对于最大的力 (5.886 N)，质量的百分比不确定度 = 0.1/600 × 100% ≈ 0.017%，与长度不确定度相比可以忽略不计。因此，误差的主要来源是伸长量的测量。

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8. Graphical Determination of Uncertainty in k | 通过图解法确定 k 的不确定度

To find the uncertainty in k, draw the best-fit line and the worst-fit lines (steepest and shallowest plausible lines through the error bars). The gradient of the steepest line gives k_max, and the shallowest gives k_min. The absolute uncertainty Δk = (k_max – k_min)/2.

为求出 k 的不确定度，绘制最佳拟合线以及最差拟合线（通过误差棒的、斜率最大和最小的合理直线）。最陡直线的斜率为 k_max，最浅的为 k_min。绝对不确定度 Δk = (k_max – k_min)/2。

If error bars are not plotted, one can estimate the uncertainty in k from the range of gradients of lines that still pass through most points.

如果没有绘制误差棒，可以通过估算仍能穿过大多数点的直线的斜率范围来估计 k 的不确定度。

For our data, using the extremes, suppose k_max ≈ 150 N m⁻¹ and k_min ≈ 142 N m⁻¹, then Δk ≈ 4 N m⁻¹, giving k = 146 ± 4 N m⁻¹. Percentage uncertainty ≈ (4/146)×100% ≈ 2.7%.

对于我们的数据，使用极端值，假设 k_max ≈ 150 N m⁻¹ 且 k_min ≈ 142 N m⁻¹，则 Δk ≈ 4 N m⁻¹，得到 k = 146 ± 4 N m⁻¹。百分比不确定度 ≈ (4/146)×100% ≈ 2.7%。

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9. Potential Sources of Error and Improvements | 潜在误差来源及改进

One common error is the misalignment of the metre rule, causing zero error. To reduce this, use a clamped rule and a small mirror behind it to avoid parallax.

一个常见误差是米尺未对准，导致零点误差。为了减少这种误差，使用夹持的米尺，并在其后放置一面小镜子以避免视差。

Another issue is that the weight of the spring itself may cause an initial extension, but this does not affect the linearity because any initial sag is constant and offsets the zero point.

另一个问题是弹簧自身的重量可能导致初始伸长，但这不影响线性关系，因为任何初始下垂是恒定的，只会偏移零点。

Repeated readings and taking an average can reduce random errors. The student could also measure the extension both when loading and unloading the masses to check for hysteresis – a sign of plastic deformation.

重复读数并取平均值可以减少随机误差。学生还可以在加载和卸载质量时测量伸长量，以检查磁滞现象——这是塑性变形的迹象。

If the spring coils touch at high loads (solid length), the spring constant appears to increase dramatically. That indicates the spring is fully compressed and the experiment should stop before this point.

如果弹簧在高负载下簧圈接触（压实长度），劲度系数会急剧增大。这表明弹簧已完全压缩，实验应在到达该点之前停止。

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10. Worked Example: Determining k from a 2018-Style Question | 案例精讲：求解 2018 风格问题中的 k

A typical exam question based on Insert 1 asks: “Using Figure 1 (force–extension graph), calculate the spring constant k. State an appropriate unit.”

基于材料1的典型考题会问：“利用图1（力–伸长图），计算弹簧劲度系数 k。写出适当的单位。”

Solution: Identify two points on the straight section: (0, 0) and (0.041 m, 5.886 N). k = ΔF/Δx = 5.886 N / 0.0410 m = 144 N m⁻¹ (approx). The answer must include the unit N m⁻¹.

解答：在直线段上选取两点：(0, 0) 和 (0.041 m, 5.886 N)。k = ΔF/Δx = 5.886 N / 0.0410 m = 144 N m⁻¹（近似）。答案必须包含单位 N m⁻¹。

Often candidates lose marks by using cm or mm without converting to metres, leading to a k expressed in N cm⁻¹ or N mm⁻¹, which is not the SI unit expected.

考生常在未将 cm 或 mm 转换为米的情况下使用它们，导致 k 用 N cm⁻¹ 或 N mm⁻¹ 表示，这不是预期的 SI 单位，因此而丢分。

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11. The Elastic Limit and Beyond | 弹性极限及其之后的行为

If the experiment continues to higher loads, the spring may undergo plastic deformation. After removing the load, the spring does not return to its original length; a permanent extension remains. This is evidence that the elastic limit has been exceeded.

如果实验继续进行到更高的负载，弹簧可能发生塑性变形。移除负载后，弹簧不会回到原长，留下不可恢复的伸长量。这证明已超过弹性极限。

In an exam, candidates might be asked to suggest how to determine the elastic limit from the graph: it is the point where the line first becomes non-linear. Alternatively, they could describe an unloading procedure and measure permanent set.

在考试中，可能会要求考生提出如何从图线确定弹性极限的方法：即线首次变为非线性的点。或者，他们可以描述卸载步骤并测量残余伸长量。

Knowledge of the difference between elastic limit and limit of proportionality is also tested. For a metal spring, these points are often nearly identical, but for other materials they can differ.

对弹性极限与比例极限之差的了解也会得到考查。对于金属弹簧，这两点通常几乎相同，但对于其他材料它们可能不同。

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12. Summary and Exam Readiness | 总结与备考提示

The June 2018 Insert 1 experiment encapsulates essential practical skills: measuring extension, tabulating data, plotting graphs, calculating gradient, and analysing uncertainties. Students should practise converting units, especially mm to m, and using the equation F = kx in context.

2018年6月材料1的实验概括了基本的实践技能：测量伸长量、将数据制成表格、绘制图线、计算斜率以及分析不确定度。学生应练习单位换算，尤其是 mm 到 m，并在情境中运用方程 F = kx。

Understanding the limitations of experimental apparatus—such as the precision of a metre rule and the effect of parallax—is as important as the mathematical treatment of data.

理解实验仪器的局限性——例如米尺的精度和视差的影响——与数据的数学处理同等重要。

By studying this reconstruction, students can better interpret unseen inserts and confidently tackle practical-based questions in their A-Level Physics examination.

通过学习这篇重建文章，学生可以更好地解读未见过的材料，并在 A-Level 物理考试中自信地应对基于实践的题目。

Published by TutorHao | Physics Revision Series | aleveler.com

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