Investigating Gravitational Acceleration | 重力加速度探究

📚 Investigating Gravitational Acceleration | 重力加速度探究

The acceleration due to gravity, denoted by \( g \), is one of the most fundamental quantities in A-Level Physics. This article explores the experimental determination of \( g \), covering theory, methods, error analysis, and exam-focused insights for CIE students.

重力加速度，用 \( g \) 表示，是 A-Level 物理中最基本的物理量之一。本文围绕 \( g \) 的实验测定展开，涵盖理论、方法、误差分析以及针对 CIE 考生的考点解析。

1. Why Measure \( g \)? | 为什么测量 \( g \)？

Knowing the value of \( g \) is essential for understanding projectile motion, pendulum oscillations, and orbital mechanics. On the Earth’s surface, the standard value is approximately \( 9.81 \, \text{m s}^{-2} \), but this varies slightly with altitude, latitude, and local geology. Measuring \( g \) accurately in the laboratory provides a hands-on test of Newton’s laws and offers insight into experimental design, data analysis, and uncertainty propagation.

知道 \( g \) 的数值，对于理解抛体运动、单摆振动和轨道力学至关重要。在地球表面，其标准值约为 \( 9.81 \, \text{m s}^{-2} \)，但该值会随海拔、纬度和局部地质条件而略有变化。在实验室中精确测量 \( g \)，既能对牛顿定律进行实操检验，也能深入理解实验设计、数据分析和不确定度传递。

2. Theoretical Foundation | 理论依据

Newton’s law of universal gravitation states that the force between a mass \( m \) and the Earth of mass \( M \) is given by \( F = \frac{GMm}{r^2} \), where \( r \) is the distance from the object to the Earth’s centre. Combining this with Newton’s second law \( F = ma \), we obtain the gravitational field strength at the Earth’s surface:

牛顿万有引力定律指出，质量为 \( m \) 的物体与质量为 \( M \) 的地球之间的引力为 \( F = \frac{GMm}{r^2} \)，其中 \( r \) 是物体到地心的距离。结合牛顿第二定律 \( F = ma \)，可得地球表面的重力场强度：

\( g = \frac{GM}{r^2} \)

Here \( G = 6.674 \times 10^{-11} \, \text{N m}^2 \, \text{kg}^{-2} \) is the gravitational constant, \( M \) is the Earth’s mass, and \( r \) is the Earth’s radius. This equation shows that \( g \) is independent of the falling object’s mass, a prediction famously verified by Galileo.

式中 \( G = 6.674 \times 10^{-11} \, \text{N m}^2 \, \text{kg}^{-2} \) 为引力常量，\( M \) 是地球质量，\( r \) 是地球半径。该式表明 \( g \) 与下落物体的质量无关，这一预言由伽利略的著名实验所验证。

3. Units and Dimensional Analysis | 单位与量纲分析

The SI unit of \( g \) is metres per second squared (\( \text{m s}^{-2} \)). From the kinematic equation \( s = \frac{1}{2} g t^2 \), we see that \( g \) has dimensions of length divided by time squared: \([g] = \text{L T}^{-2}\).

\( g \) 的 SI 单位是米每二次方秒（\( \text{m s}^{-2} \)）。由运动学方程 \( s = \frac{1}{2} g t^2 \) 可知，\( g \) 的量纲是长度除以时间的平方：\([g] = \text{L T}^{-2}\)。

A common exam trap is confusing \( g \) (acceleration) with the weight \( W = mg \) (force). Remember that \( g \) is a scalar field property, while weight is a vector force measured in newtons.

考试中常见的陷阱是混淆 \( g \)（加速度）与重力 \( W = mg \)（力）。请记住，\( g \) 是标量场属性，而重力是矢量力，单位为牛顿。

4. Method 1: Free-Fall Apparatus | 方法一：自由落体装置

The simplest laboratory method involves timing a falling ball using electromagnetic release and a trapdoor. A metal ball is held by an electromagnet; when the current is cut, a timer starts. The ball falls a distance \( s \) to a trapdoor, which stops the timer. Repeating for different heights allows \( g \) to be determined.

最简单的实验室方法是用电磁释放和落体门来计时。金属球由电磁铁吸住；切断电流时，计时器开始计时。球下落距离 \( s \) 后撞击落体门，使计时器停止。对不同高度重复实验，即可求出 \( g \)。

Using the equation \( s = \frac{1}{2} g t^2 \), a graph of \( s \) against \( t^2 \) yields a straight line through the origin with gradient \( \frac{1}{2} g \). Hence:

利用方程 \( s = \frac{1}{2} g t^2 \)，以 \( s \) 对 \( t^2 \) 作图，可得到一条过原点的直线，其斜率为 \( \frac{1}{2} g \)。因此：

\( g = 2 \times \text{gradient} \)

Key practical details: use a large distance to reduce percentage timing error, and measure \( s \) from the bottom of the ball to the trapdoor using a metre rule or vernier calliper for precision.

关键实验细节：使用较大的下落距离以减小计时的百分比误差，并用米尺或游标卡尺精确测量从球底到落体门的距离 \( s \)。

5. Method 2: Light Gates | 方法二：光电门

Light gates provide far greater timing precision than human reaction times. A card of known length \( l \) is attached to a falling mass. As the card passes through a light gate, the interruption time is recorded. The velocity \( v = \frac{l}{t} \) can be calculated for the instant the card is halfway through the gate.

光电门的计时精度远高于人的反应时间。将已知长度为 \( l \) 的挡光片固定在自由落体的重物上。当挡光片经过光电门时，记录遮光时间 \( t \)。利用 \( v = \frac{l}{t} \) 可计算挡光片经过光电门中点时刻的速度。

Placing two light gates at known separation \( h \), the equation \( v_2^2 = v_1^2 + 2 g h \) can be applied. A graph of \( v_2^2 \) against \( h \) gives a straight line of gradient \( 2g \), intercept \( v_1^2 \).

将两个光电门相距 \( h \) 放置，使用方程 \( v_2^2 = v_1^2 + 2 g h \)。绘出 \( v_2^2 \) 对 \( h \) 的图线，得到斜率为 \( 2g \)、截距为 \( v_1^2 \) 的直线。

This method eliminates the need to know the initial velocity, since it is determined from the intercept. It also reduces systematic timing errors.

该方法无需预先知道初速度，因为初速度可从截距得出，同时还能减小系统性计时误差。

6. Method 3: Simple Pendulum | 方法三：单摆

For small angular amplitudes (\( \theta < 10^\circ \)), a simple pendulum of length \( L \) has a period:

对于小角度摆动（\( \theta < 10^\circ \)），长为 \( L \) 的单摆周期为：

\( T = 2\pi \sqrt{\frac{L}{g}} \)

Squaring both sides gives \( T^2 = 4\pi^2 \frac{L}{g} \). By measuring \( T \) for different lengths \( L \), a graph of \( T^2 \) against \( L \) is a straight line with gradient \( \frac{4\pi^2}{g} \). Therefore:

两边平方得 \( T^2 = 4\pi^2 \frac{L}{g} \)。测量不同摆长 \( L \) 下的周期 \( T \)，作出 \( T^2 \) 对 \( L \) 的图线为直线，斜率 \( \frac{4\pi^2}{g} \)。因此：

\( g = \frac{4\pi^2}{\text{gradient}} \)

To reduce uncertainty, time 20 or more oscillations and divide by the number to obtain \( T \). This minimises the reaction-time error in starting and stopping the stopwatch. The length \( L \) should be measured from the pivot to the centre of the bob.

为减小不确定度，应计时 20 个或更多完整摆动，再除以次数得到 \( T \)。这能最大限度减小按停秒表时的反应时间误差。摆长 \( L \) 应从悬点量到摆球球心。

7. Data Analysis and Graphs | 数据分析与作图

Accurate graphing is crucial. When plotting, label axes with quantities and units (e.g., \( T^2 / \text{s}^2 \) on the y-axis and \( L / \text{m} \) on the x-axis). Use error bars where possible and draw the line of best fit through the plotted points.

准确作图至关重要。绘图时，必须在坐标轴上标明物理量及单位（例如 y 轴为 \( T^2 / \text{s}^2 \)，x 轴为 \( L / \text{m} \)）。尽可能加上误差棒，并画出穿过数据点的最佳拟合直线。

For the free-fall method, use \( s \) (vertical axis) against \( t^2 \) (horizontal axis). The line should pass through the origin; if it does not, this indicates a systematic error, such as a delayed timer start. The gradient is \( \frac{1}{2} g \), so remember to double the gradient value.

对于自由落体法，用 \( s \)（纵轴）对 \( t^2 \)（横轴）作图。直线应通过原点；若不过原点，则表明存在系统误差，例如计时器启动延迟。斜率为 \( \frac{1}{2} g \)，因此别忘了将斜率值乘以 2。

8. Sources of Uncertainty | 不确定度来源

The main sources of uncertainty in measuring \( g \) include:

测量 \( g \) 的主要不确定度来源包括：

Timing errors: human reaction time (about 0.2 s) or limited light-gate resolution.
Length measurement errors: metre rule precision of ±1 mm, or tape measure for large heights.
Air resistance: affects the falling object’s motion, especially at high speeds.
Systematic bias: friction in the pulley or interference from the electromagnet’s residual magnetism.

计时误差：人体反应时间（约 0.2 s）或光电门分辨率限制。
长度测量误差：米尺精度为 ±1 mm，测量较大的高度时使用卷尺。
空气阻力：影响下落物体的运动，尤其当速度较大时。
系统性偏差：滑轮摩擦或电磁铁剩磁的干扰。

For the pendulum, additional uncertainty arises from measuring the period and the exact length to the bob’s centre, as well as the approximation \( \sin \theta \approx \theta \) for small angles.

对于单摆，额外的不确定度来自周期测量、摆球球心位置的精确测量，以及小角度近似 \( \sin \theta \approx \theta \) 的近似性。

9. Improving Precision and Accuracy | 提高精密度与准确度

To obtain a reliable value of \( g \), consider these improvements:

为获得可靠的 \( g \) 值，可考虑以下改进措施：

Use light gates with data loggers to eliminate reaction time.
Repeat each measurement 3–5 times and calculate the mean.
Increase the fall height or pendulum length to reduce relative timing errors.
Use a small, dense ball to minimise air resistance.
Measure lengths with a vernier calliper or laser ruler for higher precision.

使用光电门配合数据采集器，消除反应时间。
每组数据重复测量 3–5 次并取平均值。
增加下落高度或摆长，以减小相对计时误差。
使用小而密的球，以减小空气阻力。
用游标卡尺或激光测距仪测量长度，提高精度。

In an exam, always mention that repeating and averaging reduces random error, while calibration and checking for zero error address systematic errors.

在考试中，务必提及重复测量取平均可减小随机误差，而校准仪器和检查零点误差可解决系统误差。

10. Comparing Results with the Standard Value | 与标准值比较

The accepted value of \( g \) at sea level at latitude 45° is \( 9.80665 \, \text{m s}^{-2} \). When comparing experimental results, calculate the percentage difference:

海平面纬度 45° 处的 \( g \) 标准值为 \( 9.80665 \, \text{m s}^{-2} \)。将实验结果与标准值比较时，计算百分比差异：

\( \text{Percentage difference} = \frac{|g_{\text{measured}} – g_{\text{standard}}|}{g_{\text{standard}}} \times 100\% \)

If the difference is within the experimental uncertainty, the result is consistent with the theoretical value. If not, consider systematic errors such as timing delays or incorrect length measurement.

若差异落在实验不确定度范围内，则结果与理论值一致。若不一致，则需考虑计时延迟、长度测量错误等系统误差。

The local value of \( g \) also varies: it is slightly larger at the poles than at the equator due to Earth’s rotation, and decreases with altitude. For example, at an altitude of 10 km, \( g \) is about \( 9.78 \, \text{m s}^{-2} \).

局部的 \( g \) 值也会变化：由于地球自转，两极的 \( g \) 略大于赤道，且随海拔升高而减小。例如，在 10 km 高空处，\( g \) 约为 \( 9.78 \, \text{m s}^{-2} \)。

11. Exam Tips and Common Mistakes | 答题技巧与常见错误

CIE examiners often report the following common mistakes:

CIE 考官常反馈以下常见错误：

Forgetting to double the gradient in the \( s \)-\( t^2 \) graph method.
Using the diameter instead of the radius of the pendulum bob.
Neglecting the error bars when drawing the line of best fit.
Confusing the intercept with the initial velocity in the \( v^2 \)-\( h \) graph.

在 \( s \)-\( t^2 \) 图线法中忘记将斜率乘以 2。
使用摆球直径而非半径。
绘制最佳拟合直线时忽略误差棒。
在 \( v^2 \)-\( h \) 图线中混淆截距与初速度。

Always state the units of \( g \) as \( \text{m s}^{-2} \), not \( \text{N kg}^{-1} \) unless asked specifically, since the latter is the unit of gravitational field strength, though numerically identical. In practical write-ups, explicitly quote uncertainties with the final answer, e.g., \( g = 9.8 \pm 0.2 \, \text{m s}^{-2} \).

始终将 \( g \) 的单位写为 \( \text{m s}^{-2} \)，除非题目特别要求，否则不要写成 \( \text{N kg}^{-1} \)，虽然后者是重力场强度的单位且数值相同。在实验报告中，最终结果应明确给出不确定度，例如 \( g = 9.8 \pm 0.2 \, \text{m s}^{-2} \)。

12. Summary | 总结

The investigation of gravitational acceleration is a cornerstone of experimental physics. Whether using a free-fall apparatus, light gates, or a simple pendulum, the key is to understand the underlying equations, identify sources of uncertainty, and present data through well-constructed graphs. Mastery of these methods not only secures exam marks but also builds strong practical skills for future study.

重力加速度的探究是实验物理的基石。无论是使用自由落体装置、光电门还是单摆，关键在于理解背后的方程、识别不确定度来源，并通过规范作图呈现数据。掌握这些方法，不仅能让你在考试中得分，还能为今后的学习打下扎实的实验技能基础。

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Investigating Gravitational Acceleration | 重力加速度探究