Exploring the Relationship between Period and Mass in a Spring-Mass System | 探究弹簧振子周期与质量的关系

📚 Exploring the Relationship between Period and Mass in a Spring-Mass System | 探究弹簧振子周期与质量的关系

In the Edexcel International A Level Physics Unit 5 (PH05) examination, experimental data analysis is a key skill. A classic investigation involves measuring the period of vertical oscillations of a mass-spring system to determine the spring constant and verify the relationship T = 2π√(m/k). This article provides a comprehensive walkthrough of the experimental procedure, data handling, graphical analysis, and uncertainty evaluation, mirroring the style of the June 2023 PH05 Insert.

在爱德思国际A Level物理单元5（PH05）考试中，实验数据分析是一项关键技能。一个经典的实验探究是测量弹簧振子竖直振动的周期，以确定弹簧的劲度系数并验证公式T = 2π√(m/k)。本文全面介绍了实验步骤、数据处理、图像分析和不确定度评定，其风格与2023年6月PH05插入文件中的探究任务类似。

1. Introduction to Simple Harmonic Motion and Mass-Spring System | 简谐运动与弹簧振子简介

Simple harmonic motion (SHM) occurs when the restoring force on a displaced object is directly proportional to the displacement and acts towards the equilibrium position. A mass hung on a light spring, when displaced vertically and released, executes SHM, provided the spring’s extension remains within its elastic limit. The period T of such oscillations is independent of amplitude (for small amplitudes) and is given by the well-known formula.

当物体所受的回复力与位移成正比且指向平衡位置时，物体做简谐运动（SHM）。将一个质量块挂在轻弹簧下方，竖直拉离平衡位置后释放，只要弹簧的伸长量在弹性限度内，系统就会做简谐运动。这种振动的周期T（在小振幅下）与振幅无关，并由著名的公式给出。

2. Theoretical Background | 理论背景

For an ideal massless spring, the period of oscillation T (in seconds) is related to the mass m (in kg) suspended from it and the spring constant k (in N/m) by:

对于理想的无质量弹簧，振动周期T（秒）与悬挂的质量m（千克）和弹簧劲度系数k（牛/米）的关系为：

T = 2π √(m / k)

This equation assumes that the entire mass is concentrated at the end of the spring. In practice, the spring itself has mass (mₛ), which contributes to the inertia. A more accurate expression includes an effective mass correction: T² = (4π²/k)(m + mₑff), where mₑff ≈ mₛ/3. This means a graph of T² versus m should be linear, with a gradient of 4π²/k and an intercept related to the effective spring mass.

该方程假设所有质量都集中在弹簧末端。但实际上弹簧本身也有质量（mₛ），该质量对惯性也有贡献。更精确的表达式包含有效质量修正：T² = (4π²/k)(m + mₑff)，其中mₑff ≈ mₛ/3。这意味着T²与m的图像应为一条直线，斜率为4π²/k，并且截距与弹簧的有效质量有关。

3. Experimental Set-up and Equipment | 实验装置与器材

A typical set-up includes: a helical spring (with light mass), a rigid clamp and stand, a known mass hanger, a set of slotted masses

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