📚 Determining g | 测定重力加速度 g
The acceleration due to gravity, commonly denoted by g, is one of the most fundamental constants in physics. In the laboratory, determining g to a reasonable accuracy is a classic A‑Level experiment, typically accomplished either by timing a simple pendulum or by recording the free fall of an object. Both methods rely on precise measurements of length and time, and both require careful treatment of experimental uncertainties.
重力加速度（通常用 g 表示）是物理学中最基本的常数之一。在实验室中比较准确地测定 g 是 A‑Level 物理的经典实验，一般可以通过测量单摆的周期或记录物体的自由落体来实现。这两种方法都依赖于对长度和时间进行精确测量，并且都需要谨慎处理实验误差。
1. The Acceleration Due to Gravity | 重力加速度
Near the Earth's surface, all objects in free fall accelerate downwards at approximately 9.81 m s⁻², provided air resistance is negligible. This acceleration arises from the gravitational attraction between the Earth and the object. The local value of g depends slightly on altitude, latitude and geological features, but for A‑Level purposes we aim to measure a value close to 9.8 m s⁻² in the school laboratory.
在地球表面附近，只要空气阻力可以忽略不计，所有做自由落体运动的物体都会以大约 9.81 m s⁻² 的加速度向下加速。这个加速度源于地球与物体之间的万有引力。当地的 g 值会随海拔、纬度以及地质构造的不同而略有变化，但在 A‑Level 课程中，我们在学校实验室里测得的数值往往接近 9.8 m s⁻²。
2. Principle of the Simple Pendulum | 单摆的原理
For small angular displacements (usually less than about 10°), the period T of a simple pendulum is given by
T = 2π√L/g
where L is the distance from the point of suspension to the centre of mass of the bob. Squaring both sides gives
T² = (4π²/g) L
Hence a graph of T² against L should be a straight line through the origin, with gradient = 4π²/g. From the gradient, g can be calculated as
g = 4π² / gradient
当摆角很小（通常小于 10°）时，单摆的周期 T 由下式给出：T = 2π√L/g，其中 L 是从悬点到摆球质心的距离。将两边平方可得 T² = (4π²/g) L。因此，以 T² 对 L 作图应得到一条通过原点的直线，其斜率等于 4π²/g。利用斜率可以计算出 g = 4π² / 斜率。
3. Simple Pendulum – Experimental Procedure | 单摆实验步骤
To obtain reliable data with a simple pendulum:
- Suspend a small, dense bob from a light, inextensible string about 1.0 m to 1.5 m long.
- 挂一个体积小、密度大的摆球，用轻质且几乎不可伸长的细线悬挂，线长在 1.0 m 至 1.5 m 之间。
- Measure the length L from the point of suspension to the centre of the bob using a metre rule. To increase accuracy, measure the length of the string and then add the radius of the bob.
- 用米尺测量从悬点到摆球中心的长度 L。为了提高精确度，可以先测量线长，再加上摆球的半径。
- Displace the bob by less than 10° from the vertical and release it without pushing.
- 将摆球拉离平衡位置，使摆角小于 10°，然后轻轻释放，不要施加推力。
- Using a stopwatch, time at least 20 complete oscillations (to reduce the impact of reaction time). Record the time for 20 swings, then divide by 20 to find the period T. Repeat the timing twice more and calculate the average T.
- 用秒表记录至少 20 次全振动所用的时间（以减小反应时间的影响）。用总时间除以 20 得到周期 T。重复计时两次，计算平均值 T。
- Repeat the procedure for at least six different lengths L, altering the length by adjusting the clamp.
- 改变摆长 L，至少测量六个不同长度下的周期，每次通过调节夹子来改变线长。
Throughout the experiment, ensure the pendulum swings in a single vertical plane and count oscillations carefully, using a fiduciary marker at the centre of the swing.
在整个实验过程中，应确保单摆在同一个竖直平面内摆动，并在摆动的中央位置设置一个参考标记，以便准确计数。
4. Data Analysis for the Pendulum Method | 单摆法的数据分析
Once you have pairs of values for L and T, calculate T² for each length. Plot a graph of T² (on the vertical axis) against L (on the horizontal axis). Draw a best‑fit straight line that passes through the origin. Determine the gradient of the line. Then
g = 4π² / gradient
If the line does not quite pass through the origin, a small systematic error may be present (for example, an incorrect measurement of the pendulum length). In such a case, the gradient is still used to find g, but the intercept should be discussed in the evaluation.
得到多组 L 和 T 的数据之后，计算出每个摆长对应的 T²。以 T² 为纵轴、L 为横轴作图。画出一条通过原点的最佳拟合直线，求出直线的斜率。然后根据 g = 4π² / 斜率 计算 g。如果直线没有严格通过原点，可能存在某种系统误差（例如摆长的测量有误），此时仍可用斜率计算 g，但需要在评估中讨论截距产生的原因。
The uncertainty in g can be estimated from the uncertainty in the gradient (using the difference between the maximum and minimum reasonable slopes). If the absolute uncertainty in the gradient is Δm, then the percentage uncertainty in g is the same as the percentage uncertainty in the gradient.
g 的不确定度可以根据斜率的不确定度来估算（利用最大可能斜率和最小可能斜率之差）。如果斜率的绝对不确定度为 Δm，那么 g 的百分不确定度与斜率的百分不确定度相同。
5. Principle of the Free‑Fall Method | 自由落体法的原理
An object released from rest falls a vertical distance s in time t. If the initial velocity is zero, the equation of motion is
s = ½ g t²
Thus, a graph of s against t² should be a straight line through the origin, with gradient = ½ g. Consequently,
g = 2 × gradient
This relationship holds exactly only if the object starts from rest at the instant timing begins, and if air resistance is negligible. In practice, electromagnetic release and switch‑based timing introduce small delays, so the graph may show a non‑zero intercept.
一个从静止释放的物体，在时间 t 内下落的竖直距离为 s。如果初速度为零，运动方程为 s = ½ g t²。因此，以 s 对 t² 作图应得到一条通过原点的直线，斜率等于 ½ g。于是 g = 2 × 斜率。这一关系式只有在计时开始时物体恰好由静止下落且空气阻力可忽略的情况下才严格成立。实际实验中，电磁铁释放和开关触发计时都会引入微小的延迟，因此图像可能会显示出非零截距。
6. Free‑Fall Experiment – Electromagnet and Trapdoor | 自由落体实验——电磁铁与活动门
A typical school laboratory arrangement involves:
- An electromagnet connected to a low‑voltage power supply, holding a steel ball.
- 一个连接到低压电源的电磁铁，用来吸住一枚钢球。
- A trapdoor switch (or hinged flap) placed directly below the electromagnet, connected to an electronic timer.
- 一个活动门开关（或合页板）直接放在电磁铁下方，并与电子计时器相连。
- When the current to the electromagnet is switched off, the ball falls, the timer starts, and when the ball hits the trapdoor, the timer stops.
- 断开电磁铁电流时，钢球下落，计时器开始计时；当钢球击打活动门时，计时停止。
- Measure the distance s from the bottom of the ball (when suspended) to the top of the trapdoor with a metre rule or vernier callipers.
- 用米尺或游标卡尺测量钢球悬挂时底部到活动门上表面的距离 s。
- Record the fall time t for this distance. Repeat at least three times and find the mean t.
- 记录该下落距离对应的下落时间 t。至少重复三次，计算平均时间。
- Vary the height s (e.g. 0.200 m, 0.400 m, 0.600 m, up to about 1.500 m) and repeat the measurements.
- 改变下落高度 s（例如 0.200 m, 0.400 m, 0.600 m，直至约 1.500 m），重复以上测量。
To minimise the error caused by any residual magnetism delaying the release, use a steel ball of high magnetic permeability and switch the current off sharply. The trapdoor mechanism should be sensitive and the timer must have a resolution of at least 0.001 s.
为了减小剩磁延迟释放带来的误差，应使用高磁导率的钢球，并迅速切断电流。活动门机构应足够灵敏，计时器的分辨率至少达到 0.001 s。
7. Free‑Fall Data Analysis | 自由落体数据分析
For each height s calculate t², where t is the average fall time. Plot a graph of s (vertical axis) against t² (horizontal axis). Draw the best‑fit straight line; the gradient m should be ½ g. Hence
g = 2 m
If the line intercepts the t² axis positively, this indicates that a small time delay occurred before the timer started – for instance, because the ball took a tiny but finite time to leave the electromagnet after the current was cut. The intercept can be discussed in the evaluation, but the gradient still yields a reliable g.
对每一个下落高度 s，计算 t²（其中 t 是平均下落时间）。以 s 为纵轴、t² 为横轴作图。画出最佳拟合直线，其斜率 m 应等于 ½ g。因此 g = 2 m。如果直线与 t² 轴正方向有截距，说明在计时器启动之前存在一段微小的时间延迟——例如，切断电流后钢球需要一段极短但不可忽略的时间才能脱离电磁铁。可以在评估中讨论这一截距，但斜率仍然能给出可靠的 g 值。
Estimate the uncertainty in g by finding the maximum and minimum plausible gradients (steepest and shallowest lines that still fit the data points within their error bars). Half the difference gives the absolute uncertainty in the gradient, and this percentage uncertainty propagates directly to g.
估算 g 的不确定度时，可以找出最大可能的斜率和最小可能的斜率（即仍然能合理拟合误差棒的最陡和最平缓的直线）。两者的差值一半即为斜率的绝对不确定度，该百分不确定度直接传递给 g。
8. Free‑Fall Using Two Light Gates | 使用双光电门的自由落体法
An alternative free‑fall method avoids the problem of delayed release by using two light gates connected to a data logger. A card of known length interrupts each beam in turn, allowing the logger to record the velocities v₁ and v₂ at the two gates, and the time t taken to travel between them. Using the equation
v₂ = v₁ + g t
a graph of v against t (for several positions of the second gate) yields a straight line of gradient g. This method is independent of the starting conditions and often gives more precise results.
另一种自由落体方法利用两个光电门连接数据采集器，从而避免了释放延迟的问题。一个已知长度的挡光片先后遮断两道光束，采集器可记录下小球通过两个光电门时的速度 v₁ 和 v₂，以及两者之间的运动时间 t。根据 v₂ = v₁ + g t，以 v 对 t 作图（改变第二个光电门的位置获得多组数据），可得到一条斜率为 g 的直线。这种方法与起始条件无关，通常能给出更精确的结果。
9. Sources of Uncertainty and Errors | 误差来源
Both pendulum and free‑fall experiments share common sources of uncertainty:
- Parallax error when measuring lengths with a rule; this can be reduced by using a set‑square or by sighting at eye level.
- 用直尺测量长度时的视差误差；可通过使用三角尺或保持视线水平来减小。
- Reaction time in starting and stopping a stopwatch; using a large number of oscillations or electronic timing minimises this.
- 启动和停止秒表时的反应时间；通过增加摆动次数或采用电子计时可将这一误差降至最低。
- The assumption of small‑angle oscillations in the pendulum; keeping the amplitude below 10° keeps the error in T below about 0.5 %.
- 单摆中小角度摆动的假设；将振幅控制在 10° 以下可使周期误差不超过约 0.5 %。
- Air resistance affects both methods but is more significant for light bobs or objects with large surface areas; using small, dense spheres reduces drag.
- 空气阻力对两种方法都有影响，但对轻质摆球或表面积大的物体更为显著；使用体积小、密度大的球体可以减小阻力。
- Residual magnetism in the electromagnet causing delayed release; clean contacts and a sharp cut‑off are important.
- 电磁铁的剩磁导致释放延迟；确保触点清洁并瞬间切断电流很重要。
- The trapdoor’s mechanical response time; verified by checking that the timer starts and stops promptly.
- 活动门的机械响应时间；可通过检查计时器启停是否及时来验证。
Systematic errors (such as a mis‑calibrated timer or an incorrectly measured length) shift all readings in the same direction and are best detected by plotting a graph and examining the intercept.
系统误差（如计时器校准不准或长度测量错误）会使所有读数朝同一方向偏移，通过作图并观察截距可以很好地发现此类误差。
10. Comparison of Methods and Exam Tips | 方法比较与考试技巧
In CIE A‑Level examinations, you may be asked to describe either method in detail or to comment on an experimental set‑up. Remember:
- The pendulum method requires a long string, small amplitudes, and timing of many oscillations; the derived quantity is T², which is plotted against L.
- 单摆法需要长的摆线、小摆角并记录多次摆动的时间；导出的量是 T²，并对 L 作图。
- The free‑fall method (electromagnet plus trapdoor) uses s = ½ g t² and a plot of s vs t².
- 自由落体法（电磁铁加活动门）利用 s = ½ g t²，并作 s–t² 图。
- In both cases, a straight‑line graph confirms the theory and allows g to be found from the gradient. State clearly how the gradient relates to g.
- 两种情况下，一条过原点或近似过原点的直线图形都能验证理论，并能通过斜率求出 g。务必清楚说明斜率与 g 的数学关系。
- Mark schemes reward precise language: ‘plot a graph of T² against L’, ‘gradient = 4π²/g’, ‘repeat and average’, ‘use a fiduciary marker’, ‘small amplitude (<10°)’, ‘use a set‑square to avoid parallax’, etc.
- 评分标准看重准确的表述：“作 T² 对 L 的图”，“斜率 = 4π²/g”，“重复测量取平均值”，“使用基准标记”，“小幅度（<10°）”，“使用三角尺避免视差”等。
When discussing errors, distinguish clearly between random and systematic effects, and suggest practical improvements such as using longer distances, digital timers, or light gates.
在讨论误差时，要清晰地区分随机效应与系统效应，并提出切实可行的改进措施，例如使用更长的下落距离、数字计时器或光电门。
Published by TutorHao | Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图