Investigating How Different Temperatures Affect the Spring Constant of a Spring | 探究不同温度对弹簧劲度系数的影响

📚 Investigating How Different Temperatures Affect the Spring Constant of a Spring | 探究不同温度对弹簧劲度系数的影响

The spring constant is a fundamental parameter in Hooke’s law, describing the stiffness of a spring. While often treated as a constant, it can vary with temperature due to changes in the material’s elastic modulus. This investigation explores how different temperatures affect the spring constant of a metal coil spring, applying controlled heating and cooling methods and analyzing the results with uncertainty considerations. Such an experiment aligns with the IB Physics internal assessment, emphasizing practical skills and data analysis.

弹簧常数是胡克定律中的一个基本参数，描述弹簧的刚度。虽然常被视为常数，但由于材料弹性模量的变化，它会随温度而变化。本探究利用受控加热与冷却方法，研究不同温度如何影响金属螺旋弹簧的劲度系数，并结合不确定度分析结果。该实验与 IB 物理内部评估要求一致，着重于实践技能和数据分析。

1. Hooke’s Law and the Spring Constant | 胡克定律与弹簧常数

Hooke’s law states that the force F applied to extend or compress a spring is directly proportional to the displacement x from its equilibrium position.

F = k x

胡克定律指出，拉伸或压缩弹簧的力 F 与其偏离平衡位置的位移 x 成正比，比例系数 k 即为弹簧常数（劲度系数），单位为 N m⁻¹。k 越大，弹簧越硬。对于理想的线性弹簧，k 在弹性限度内被看作常数，但实际上它依赖于弹簧的材料和几何形状。

For a helical spring, the spring constant can be expressed in terms of the material’s shear modulus G and the spring’s geometry:

k = (G d⁴) / (8 D³ N)

其中 d 是线径，D 是平均螺旋直径，N 是有效圈数。这个关系式表明，任何改变剪切模量 G 的因素（如温度）都会直接导致 k 的变化。因此，探究温度对弹簧常数的影响实质上是在分析材料的弹性模量如何随温度改变。

2. Temperature Dependence of Elastic Modulus | 弹性模量的温度依赖性

The shear modulus G of most metals decreases as temperature rises, primarily because increased atomic vibrations weaken the interatomic bonds. A simple linear model describes this behaviour:

G(T) = G₀ (1 – α ΔT) or G(T) = G₀ (1 – β (T – T₀))

其中 G₀ 是参考温度 T₀（常取 20 °C 或 0 °C）下的剪切模量，α（或 β）为材料的正温度系数，单位 K⁻¹。对于大多数金属，α 约为 10⁻⁴ K⁻¹ 量级。例如，钢的剪切模量温度系数大致为 0.03% per °C。当温度升高 ΔT > 0，G(T) < G₀，导致 k 也相应减小。冷却时弹性模量上升，弹簧变硬。这一理论预测正是本实验要验证的核心。

3. Aim, Hypothesis and Variables | 实验目的、假设与变量

The aim of this investigation is to determine how the spring constant k of a metal coil spring changes with temperature, using a static loading method. The hypothesis is that increasing temperature will cause a measurable decrease in k, while lowering temperature will increase k, consistent with the thermal softening of the shear modulus.

实验目的是通过静态加载法确定金属螺旋弹簧的劲度系数 k 如何随温度变化。假设升温会导致 k 明显下降，降温会使 k 增大，这与剪切模量的热软化行为一致。实验中的自变量是弹簧所处的水浴温度 T，因变量为弹簧常数 k。控制变量包括：弹簧本身（同一根弹簧、同一段有效圈数），每次加载的砝码质量范围（避免塑性变形），测量伸长量的方法（直尺位置固定），以及环境振动等。保持这些因素不变才能确保观察到的 k 变化仅来自温度效应。

4. Apparatus and Experimental Setup | 实验器材与装置

The experiment requires the following equipment: a metal helical spring (e.g. steel or brass), a set of slotted masses (50 g to 250 g), a metre ruler with millimetre divisions, a thermometer capable of reading from -10 °C to 110 °C, a water bath or large beaker, an electric heater or hot plate, ice, a retort stand with clamp, and a vernier caliper to measure the spring’s wire and coil diameters. Safety precautions include handling hot water with tongs and wearing eye protection.

实验所需器材包括：一根金属螺旋弹簧（如钢或黄铜）、一套槽码（50 g 至 250 g）、毫米刻度直尺、量程为 -10 °C 至 110 °C 的温度计、水浴锅或大烧杯、电加热器或加热板、冰块、铁架台及夹具、游标卡尺（用于测量线径和螺旋直径）。安全措施包括用夹钳处理热水、佩戴护目镜。

In the setup, the spring is hung vertically from the clamp, with the lower end free to attach masses. The water bath is raised so that the entire spring is submerged. The thermometer is placed close to the spring to monitor the water temperature. A loading mass hanger is attached, and the metre ruler is fixed vertically alongside the spring, with its zero aligned to the unstretched lower end of the spring (using a fiducial marker to reduce parallax). Before taking data, the system must reach thermal equilibrium—typically 5 minutes after the water bath reaches the target temperature.

实验装置中，弹簧竖直悬挂于夹具，下端可挂砝码架。水浴容器升至弹簧完全浸没的位置，温度计靠近弹簧监测水温。在弹簧旁边固定一把垂直米尺，尺的零点对准弹簧未拉伸时下端的位置（使用指针标记以减小视差）。加载砝码前，必须等待水温达到目标值并稳定约 5 分钟，确保弹簧与水浴热平衡。

5. Procedure | 实验步骤

1. Measure the spring’s wire diameter d and coil diameter D using the vernier caliper; calculate the expected room-temperature k using the formula from Section 1 if desired. Record the room temperature T₀. Suspend the spring without any load and record the initial position of the pointer on the ruler (L₀). Add masses in steps of 50 g up to 250 g, recording the new pointer position L each time. The extension is x = L – L₀. For each mass, calculate the force F = m g (using g = 9.81 N kg⁻¹) and the spring constant k = F / x. Repeat the loading-unloading cycle twice to check for hysteresis and obtain an average k at room temperature.

1. 用游标卡尺测量弹簧线径 d 和螺旋直径 D；如需可用前述公式估算室温下的 k。记录室温 T₀。不加载任何砝码，将弹簧悬挂并记录指针在尺上的初始位置 L₀。依次增加 50 g 砝码直至 250 g，每次记录指针位置 L。伸长量 x = L – L₀。对每个质量，计算力 F = m g（g =

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