📚 Investigation of Free Fall to Determine g | 自由落体实验探究：测定重力加速度g
The acceleration due to gravity, g, is a fundamental constant that describes the rate at which objects accelerate towards the Earth when dropped from rest. In the laboratory, it is often determined using a free-fall method involving an electromagnet, a steel ball and a trapdoor switch connected to an electronic timer. This experiment provides an excellent opportunity to apply kinematic equations, analyse uncertainties and evaluate systematic errors.
重力加速度g是一个基本常数，描述物体从静止开始下落时朝向地球加速的速率。实验室中通常采用自由落体方法测定g，使用电磁铁、钢球和连接电子计时器的落体开关。该实验为应用运动学方程、分析不确定度并评估系统误差提供了极好的机会。
1. Aim | 实验目的
To measure the acceleration due to gravity, g, by timing the free fall of a steel ball over a measured vertical distance using an electromagnet and trapdoor timer, and to evaluate the reliability of the result by considering sources of uncertainty and systematic error.
通过使用电磁铁和落体计时器测量钢球在已知竖直距离上自由下落的时间，测定重力加速度g，并通过考虑不确定度来源和系统误差来评估结果的可靠性。
2. Apparatus | 实验器材
The apparatus includes a low-voltage electromagnet to hold and release the steel ball without imparting initial velocity, a steel ball bearing, a trapdoor switch that opens instantly when struck by the ball, an electronic timer (resolution 0.01 s or better) that starts when the electromagnet circuit is broken and stops when the trapdoor opens, a metre ruler (resolution 1 mm), a plumb line to ensure vertical alignment, a clamp and stand, and connecting leads.
实验器材包括低压电磁铁（用于吸住并释放钢球且不施加初速度）、钢球、钢球撞击时瞬间打开的落体开关、电子计时器（分辨率0.01秒或更佳，在电磁铁电路断开时启动并在落体开关打开时停止）、米尺（分辨率1 mm）、用于确保竖直对齐的铅垂线、铁架台和连接导线。
The timer must be capable of measuring short time intervals with minimal reaction-time error. A digital timer triggered by electrical contacts is preferred over a stopwatch, as human reaction time would dominate the uncertainty.
计时器必须能够以最小的反应时间误差测量短时间间隔。由电触点触发的数字计时器优于秒表，因为人为反应时间会成为不确定度的主要来源。
3. Variables | 变量
Independent variable: The vertical drop height h, measured from the bottom of the ball when held by the electromagnet to the top surface of the trapdoor. This distance is varied in suitable increments, e.g. 0.200 m, 0.400 m, 0.600 m, up to about 1.000 m.
自变量：竖直下落高度h，从电磁铁吸住时球的底部到落体开关上表面的距离。该距离以适当增量变化，例如0.200 m、0.400 m、0.600 m，最高约1.000 m。
Dependent variable: The time t taken for the ball to fall from release until the trapdoor opens. Several repeat readings are taken at each height to minimise random error, and the mean time is calculated.
因变量：球从释放到落体开关打开所需的下降时间t。在每个高度处进行多次重复读数以减小随机误差，并计算平均时间。
Control variables: The same steel ball is used throughout to keep mass and size constant. The electromagnet current is kept unchanged to ensure a clean release without delay. The apparatus is shielded from draughts, and the release is performed gently to avoid giving the ball an initial downward push. The trapdoor switch is not moved, so its position defines a fixed reference plane.
控制变量：全程使用同一钢球以保持质量和大小不变。电磁铁电流保持不变，确保释放干脆无延迟。实验装置避风，且释放操作轻柔，避免给球施加向下的初速度。落体开关位置固定不动，以定义不变的参考面。
4. Procedure | 实验步骤
1. Assemble the apparatus so that the electromagnet is clamped vertically above the trapdoor. Use the plumb line to check that the ball will fall centrally through the trapdoor without touching the sides.
1. 组装仪器，使电磁铁垂直固定在落体开关上方。用铅垂线检查钢球能否居中通过落体开关而不触碰侧面。
2. Set the height h to the first value by measuring with the metre ruler from the bottom of the suspended ball to the trapdoor surface. Record the height.
2. 用米尺从悬挂钢球的底部测量至落体开关表面的距离，将高度h设定为第一个值，并记录高度。
3. Switch on the electromagnet and attach the steel ball. Reset the timer to zero. Switch off the electromagnet; the timer starts and the ball falls. As soon as the ball hits the trapdoor, the timer stops. Record the time t.
3. 接通电磁铁并吸住钢球。计时器复位为零。断开电磁铁；计时器启动，钢球下落。钢球一撞击落体开关，计时器立刻停止。记录时间t。
4. Repeat the measurement at least three times for this height to obtain a mean time. Discard any anomalous results caused by the ball touching the sides or the trapdoor not triggering cleanly.
4. 在此高度下重复测量至少三次以获得平均时间。剔除由球碰壁或落体开关未干净触发所致的异常结果。
5. Increase the height by releasing the clamp and resetting it to the next value. Repeat the timing procedure for each height across the planned range.
5. 松开铁夹并重新设定到下一个高度值，增大高度。对计划范围内的每个高度重复计时过程。
5. Data Collection | 数据收集
Organise the data in a table with columns: h (m), t₁ (s), t₂ (s), t₃ (s), mean t (s), and t² (s²). Record the height to the nearest millimetre (0.001 m) and the timer readings to two or three decimal places as appropriate.
将数据整理在表格中，列包括：h (m)、t₁ (s)、t₂ (s)、t₃ (s)、平均t (s) 和t² (s²)。高度记录到毫米（0.001 m），计时器读数视情况记录到小数点后两位或三位。
| h / m | t₁ / s | t₂ / s | t₃ / s | mean t / s | t² / s² |
|---|---|---|---|---|---|
| 0.200 | 0.204 | 0.202 | 0.203 | 0.203 | 0.0412 |
| 0.400 | 0.286 | 0.288 | 0.287 | 0.287 | 0.0824 |
Calculate t² for each mean time. The squared time will be used in the graphical analysis because the kinematic equation predicts a linear relationship between h and t².
计算每个平均时间的t²。由于运动学方程预测h与t²之间存在线性关系，平方时间将用于图像分析。
6. Data Analysis and Graph | 数据分析与图像
For an object starting from rest with uniform acceleration g, the distance fallen is given by:
h = ½ g t²
对于从静止开始以匀加速度g下落的物体，下落距离由下式给出：
h = ½ g t²
Plot a graph of h (vertical axis) against t² (horizontal axis). If the free-fall model holds, the points should lie on a straight line passing through the origin. The gradient of this line is equal to ½ g. Therefore, g can be determined as:
g = 2 × gradient
绘制h（纵轴）对t²（横轴）的图。如果自由落体模型成立，数据点应位于一条通过原点的直线上。该直线的梯度等于½ g，因此g可由下式求得：
g = 2 × 梯度
Use a best-fit line and calculate the gradient from a large triangle on the graph. An alternative approach is to calculate g for each data pair using g = 2h / t² and then determine the mean value, but the graphical method is preferable because it averages out random errors and allows the intercept to be checked.
使用最佳拟合线，从图像上的大三角形计算梯度。另一种方法是利用g = 2h / t²计算每对数据的g值再求平均，但图像法更可取，因为它能平均随机误差并可检验截距。
7. Uncertainty Calculation | 不确定度计算
The metre ruler has a scale division of 1 mm, giving an absolute uncertainty of ±0.001 m for each height measurement. However, the difficulty in aligning the ball and the trapdoor means the practical uncertainty in h is often about ±0.002 m. The percentage uncertainty in h for the smallest height (0.200 m) is:
%U(h) = (0.002 / 0.200) × 100% = 1.0%
米尺的最小分度为1 mm，每次高度测量的绝对不确定度为±0.001 m。然而，对齐钢球与落体开关的困难意味着h的实际不确定度约为±0.002 m。在最小高度（0.200 m）下，h的百分不确定度为：
%U(h) = (0.002 / 0.200) × 100% = 1.0%
The timer uncertainty is typically the resolution of the digital display, e.g. ±0.005 s if the timer reads to 0.01 s. For a short time like 0.203 s, the percentage uncertainty is:
%U(t) = (0.005 / 0.203) × 100% ≈ 2.5%
计时器的不确定度通常为数字显示的分辨率，例如若计时器读到0.01 s，则为±0.005 s。对于0.203 s这样短的时间，百分不确定度为：
%U(t) = (0.005 / 0.203) × 100% ≈ 2.5%
Since t is squared, the percentage uncertainty in t² is approximately 2 × %U(t) = 5.0%. The combined random uncertainty in the gradient can be estimated by drawing worst-fit lines (steepest and shallowest lines that still pass through the error bars) and finding the range of gradients. This gives an uncertainty in g.
由于t被平方，t²的百分不确定度大约为2 × %U(t) = 5.0%。梯度中的综合随机不确定度可通过绘制最差拟合线（仍通过误差棒的陡峭和最平缓直线）并求出梯度范围来估计，从而得出g的不确定度。
8. Sources of Error | 误差来源
Systematic error from electromagnet delay: When the electromagnet is switched off, residual magnetism may cause the ball to be released a fraction of a second after the circuit breaks. This makes the measured time longer than the true free-fall time, leading to a lower calculated value of g.
电磁铁延迟导致的系统误差：断开电磁铁时，剩磁可能导致钢球在电路断开后一瞬间才释放，这使得测量时间比真实自由落体时间长，从而计算出较小的g值。
Air resistance: At higher speeds, air resistance becomes noticeable and reduces the acceleration, again causing g to be underestimated. This effect is more significant for larger heights and lighter balls.
空气阻力：在较高速度下，空气阻力变得明显并减小加速度，同样导致g被低估。此效应对较大高度和较轻球体更为显著。
Parallax error in height measurement: Viewing the metre ruler from an angle when aligning the bottom of the ball introduces a random error in h.
高度测量中的视差：对齐钢球底部时，若从某一角度看米尺，会给h带来随机误差。
Non-zero initial velocity: If the release is not perfect, the ball may acquire a tiny downward velocity, which would reduce the measured time slightly and produce an overestimate of g.
非零初速度：若释放不完美，球可能获得微小的向下速度，这将略微缩短测量时间，并高估g。
9. Improvements | 改进措施
Replace the electromagnet-trapdoor system with a pair of light gates connected to a data logger. When the ball passes the first gate, timing starts; when it passes the second, timing stops. This eliminates the release delay and the mechanical switch bounce of the trapdoor, significantly reducing systematic error.
用连接数据记录器的一对光门替代电磁铁-落体开关系统。当钢球经过第一道光门时开始计时，经过第二道时停止计时。这消除了释放延迟和落体开关的机械弹跳，显著减小系统误差。
Use a larger, denser steel ball to reduce the effect of air resistance, or conduct the experiment in a vacuum chamber to eliminate air drag altogether. The vacuum method gives the most accurate determination of g.
使用更大、更致密的钢球以减小空气阻力效应，或在真空室中进行实验以完全消除空气阻力。真空法能给出最准确的g值。
To minimise parallax, use a set square to align the bottom of the ball with the scale, or mount a small pointer on the clamp that indicates the reference point on the ruler. Repeat the experiment with a range of heights covering at least a factor of 5 to improve the reliability of the gradient.
为减小视差，使用直角尺将球底部与刻度对齐，或在铁夹上安装一个小指针指示米尺上的参考点。选用至少5倍范围的高度重复实验，以提高梯度的可靠性。
10. Conclusion | 结论
The experimental value of g obtained from the gradient is typically within the range 9.5 – 9.9 m s⁻² if systematic delays are present. With careful technique and the improvements suggested, a value much closer to the accepted 9.81 m s⁻² can be achieved. The experiment demonstrates the power of linearising a quadratic relationship, assessing uncertainties both graphically and through calculation, and critically evaluating the impact of systematic errors on the final result.
若存在系统延迟，通过梯度得到的g实验值通常在9.5–9.9 m s⁻²范围内。凭借仔细操作和上述改进，可以得到更接近公认值9.81 m s⁻²的结果。该实验展示了将二次关系线性化、通过图像和计算评估不确定度、以及批判性评估系统误差对最终结果影响的能力。
The investigation reinforces the understanding that g can be determined from a simple free-fall setup, but the accuracy hinges on minimising electromagnetic delay, air effects and timing uncertainties. It also highlights the importance of repeat measurements and appropriate graph plotting in reducing random error and revealing systematic trends.
本探究强化了如下认识：g可通过简单的自由落体装置测定，但其准确性取决于电磁延迟、空气效应和计时不确定度的最小化。它还突显了重复测量和恰当绘图在减小随机误差、揭示系统趋势中的重要性。
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