📚 Experimental Investigation of Circular and Periodic Motion | 圆周运动与周期运动实验探究
In A-Level Physics, exploring circular and periodic motion through hands-on experiments is essential for developing a deep understanding of core concepts like centripetal force, angular velocity, and simple harmonic motion. This practical investigation brings together two interconnected domains—uniform circular motion and oscillatory systems such as mass-spring and pendulum setups—to help students verify theoretical relationships, refine measurement skills, and appreciate the importance of controlled variables. By systematically measuring force, period, mass, radius, and length, one can confirm the predictions of Newton’s laws and Hooke’s law while gaining insight into real-world applications ranging from satellite orbits to mechanical vibrations.
在A-Level物理中，通过动手实验探究圆周运动和周期运动，对于深刻理解向心力、角速度和简谐运动等核心概念至关重要。本次实验探究将匀速圆周运动与弹簧振子和单摆等振动系统这两个相互关联的领域结合起来，帮助学生验证理论关系、提升测量技能，并认识到控制变量的重要性。通过系统地测量力、周期、质量、半径和摆长，可以验证牛顿定律和胡克定律的预言，同时深入认识从卫星轨道到机械振动的实际应用。
1. Aims of the Experiment | 实验目的
The primary aim is to investigate the relationship between centripetal force, mass, radius, and angular velocity for an object in uniform circular motion, and to explore the factors affecting the period of simple harmonic oscillators such as a mass-spring system and a simple pendulum. By collecting and analysing data, students will determine whether experimental results align with the theoretical equations F = m r ω², T = 2π √(m/k) and T = 2π √(L/g).
主要目的是探究作匀速圆周运动的物体其向心力与质量、半径和角速度之间的关系，并探究影响弹簧振子与单摆这类简谐振子周期的因素。通过收集和分析数据，学生将判断实验结果是否符合理论方程 F = m r ω²、T = 2π √(m/k) 以及 T = 2π √(L/g)。
2. Theoretical Background | 理论背景
An object moving in a circle of radius r with constant speed v has a centripetal acceleration a = v²/r = r ω², where ω is the angular speed. According to Newton’s second law, the net inward force required is F = m a = m r ω². In periodic motion, a mass attached to a spring obeys Hooke’s law F = –k x, resulting in a period T = 2π √(m/k) for small amplitudes. For a simple pendulum of length L, the period for small angles (θ < 10°) is independent of mass and amplitude, given by T = 2π √(L/g), where g is the gravitational field strength.
一个物体以恒定速率 v 在半径为 r 的圆周上运动时，其向心加速度为 a = v²/r = r ω²，其中 ω 为角速度。根据牛顿第二定律，所需的净向心力为 F = m a = m r ω²。在周期运动中，连接在弹簧上的重物遵循胡克定律 F = –k x，在小振幅下其周期 T = 2π √(m/k)。对于摆长为 L 的单摆，在小角度（θ < 10°）下周期与质量和振幅无关，由 T = 2π √(L/g) 给出，其中 g 为重力场强度。
3. Apparatus and Setup for Circular Motion | 圆周运动实验装置
The circular motion experiment typically uses a whirling bung on a string, a vertical shaft with a mass carrier, or a purpose-built centripetal force apparatus. A reliable setup includes a rubber bung of known mass m, a strong nylon string threaded through a glass or plastic tube, and a slotted mass hanger attached to the lower end of the string. The tube is held vertically; when the bung is whirled in a horizontal circle, the hanging weight Mg provides the centripetal force, assuming negligible friction at the tube rim. A marker clip is fixed on the string below the tube to maintain a constant radius.
圆周运动实验通常使用旋转橡胶塞、带质量支架的立轴或专用向心力实验仪。一种可靠的装置包括已知质量 m 的橡胶塞、穿过玻璃或塑料管的坚固尼龙线，以及悬挂在绳下端的槽码挂钩。垂直握住管子；当橡胶塞在水平面内旋转时，悬挂重量 Mg 提供向心力，前提是管口摩擦可忽略。在管下方绳子上固定一个标志夹子，用以保持半径恒定。
| Apparatus | Quantity/ Specification |
| Rubber bung with hole | 1 (mass ~ 0.05 kg) |
| Nylon string (1.5 m) | 1 |
| Glass/plastic tube (smooth ends) | 1 |
| Slotted masses & hanger | set up to 200 g |
| Marker clip or tape | 1 |
| Stopwatch | ±0.01 s |
| Metre rule | ±1 mm |
| Digital balance | ±0.1 g |
4. Procedure for Circular Motion | 圆周运动实验步骤
Measure the mass m of the rubber bung using the digital balance. Attach the bung securely to one end of the string. Pass the free end through the tube and attach the mass hanger. Add slotted masses to achieve a hanging mass M (e.g., 100 g). Adjust the marker clip so that the radius r from the top of the tube to the centre of the bung is a chosen value, say 0.60 m. Whirl the bung in a horizontal circle with a steady motion so that the marker clip stays just below the bottom of the tube without touching it. Once steady, use the stopwatch to time 20 complete revolutions; record the time. Repeat for the same M and r at least three times to obtain an average period T_rev. Vary the hanging mass M (altering centripetal force), or vary the radius r, or change the bung mass, while keeping other variables constant in each investigation.
用数字天平测量橡胶塞的质量 m。将橡胶塞牢牢系在绳子一端。将绳子自由端穿过玻璃管，系上质量挂钩。添加槽码，使悬挂质量 M（例如 100 g）。调整标志夹子，使管口到橡胶塞中心的半径 r 为选定值，例如 0.60 m。在水平面内匀速旋转橡胶塞，使标志夹子始终保持在管底下方但不接触。稳定后，用秒表测量 20 次完整旋转所需的时间；记录时间。对同一组 M 和 r 重复至少三次，得到平均周期 T_rev。分别改变悬挂质量 M（改变向心力）、改变半径 r 或改变橡胶塞质量，每次探究中保持其他变量不变。
5. Data Analysis: Circular Motion | 圆周运动数据分析
For each trial, calculate the period of one revolution T = t/20, then the angular speed ω = 2π/T. The theoretical centripetal force supplied by the hanging weight is F_theor = M g, where g = 9.81 m s⁻². The experimentally required centripetal force can be computed as F_exp = m r ω². Compare F_theor and F_exp; ideally they should agree within experimental uncertainty. Plot a graph of F_theor against ω² for constant m and r; a straight line through the origin is expected with gradient equal to m r. Alternatively, keep M and m constant and plot F against r; the slope should equal m ω². Calculate percentage differences and comment on sources of error.
每次试验计算一次旋转的周期 T = t/20，然后计算角速度 ω = 2π/T。悬挂重物提供的理论向心力为 F_theor = M g，其中 g = 9.81 m s⁻²。实验所需的向心力可计算为 F_exp = m r ω²。比较 F_theor 和 F_exp；理想情况下两者应在实验误差范围内一致。对固定的 m 和 r，绘制 F_theor 对 ω² 的图线；预期为一条过原点的直线，斜率等于 m r。或者保持 M 和 m 不变，绘制 F 对 r 的图线，斜率应等于 m ω²。计算百分比差异并评述误差来源。
F_theor = M g and F_exp = m r ω²
6. Investigating Simple Harmonic Motion: Spring-Mass System | 简谐运动实验：弹簧振子
Suspend a spring of negligible mass from a rigid support. Attach a slotted mass carrier and add masses to achieve a total oscillating mass m_s. Displace the mass vertically by a small amount (≤ 5 cm) and release. Measure the time for 20 complete oscillations; determine the period T. Repeat for at least six different masses, keeping amplitude small. According to theory, T² = (4π²/k) m_s, so a graph of T² against m_s should be a straight line passing through the origin (if the spring’s effective mass is negligible). The spring constant k can be calculated from the slope: k = 4π² / slope. Additionally, verify that T does not depend on amplitude for small displacements.
将质量可忽略的弹簧悬挂在刚性支架上。挂上槽码支架并添加质量，得到总振动质量 m_s。将质量块向下拉伸一小段距离（≤ 5 cm）后释放。测量 20 次完整振动的时间，确定周期 T。对至少六种不同质量重复实验，保持振幅较小。根据理论，T² = (4π²/k) m_s，因此 T² 对 m_s 的图线应为过原点的直线（若弹簧有效质量可忽略）。弹簧劲度系数 k 可由斜率求得：k = 4π² / 斜率。此外，验证在小位移下周期 T 不依赖于振幅。
7. Investigating Simple Harmonic Motion: Simple Pendulum | 简谐运动实验：单摆
Clamp a small, dense spherical bob to a light, inextensible string. Measure the length L from the point of suspension to the centre of the bob. Displace the bob by a small angle (< 10°) and release. Time 20 oscillations and find the period T. Repeat for lengths ranging from 0.30 m to 1.20 m. Plot a graph of T² against L; a straight line through the origin should be obtained with gradient 4π²/g. Determine the experimental value of g from the slope and compare with the accepted value of 9.81 m s⁻². Check that changing the mass of the bob (using different bobs of the same size) does not alter T for the same length, confirming mass independence.
用夹子将一个体积小、密度大的球形摆球固定在质量轻且不可伸长的细绳上。测量从悬点到摆球中心的摆长 L。使摆球偏离平衡位置一个小角度（< 10°）后释放。测量 20 次振动的时间，求出周期 T。对从 0.30 m 到 1.20 m 的不同摆长重复实验。绘制 T² 对 L 的图线；应得到一条过原点的直线，斜率为 4π²/g。由斜率计算出实验的 g 值，并与公认值 9.81 m s⁻² 进行比较。验证改变摆球质量（使用相同大小但不同材料的摆球）不会改变相同摆长下的周期 T，从而确认质量无关性。
T = 2π √(L/g) → T² = (4π²/g) L
8. Error Analysis and Improvements | 误差分析与改进
Common systematic errors in the circular motion experiment include friction at the tube rim, which reduces the actual tension transmitted to the bung, and the string not being exactly horizontal, introducing a vertical component. Random errors arise from difficulty in keeping the radius exactly constant while whirling and from human reaction time in stopwatch measurements. To reduce these, use a low-friction tube (lubricated or polished), employ a reference pointer to monitor radius, and time a larger number of revolutions (e.g., 30 or 40). For spring and pendulum experiments, ensure oscillations are strictly in one plane, use a fiducial marker at the equilibrium position for accurate timing, and measure length with vernier callipers to improve precision. Always take repeat readings and calculate mean periods; use graphical methods to minimise the effect of outliers.
圆周运动实验中的常见系统误差包括管口摩擦，这会降低传递给橡胶塞的实际张力，以及绳子并非严格水平而引入了垂直分量。随机误差源于旋转时难以保持半径精确恒定，以及秒表测量中的人为反应时间。为减小误差，可使用低摩擦管（润滑或抛光）、采用参考指示器来监测半径，并测量更多次旋转的时间（如 30 或 40 次）。对于弹簧和单摆实验，确保振动严格在同一平面内，在平衡位置安放基准标记以便准确计时，并使用游标卡尺测量长度以提高精度。始终进行重复读数并计算平均周期；采用图像法以减小异常值的影响。
9. Safety Precautions | 安全注意事项
When conducting the whirling bung experiment, wear safety goggles in case the string snaps. Ensure the area is clear of other students, and do not whirl the bung at excessive speed. The hanging masses should be securely attached, and a soft landing surface (e.g., a foam mat) should be placed underneath to catch falling weights. For the spring-mass system, avoid overstretching the spring beyond its elastic limit; use a support stand with a heavy base to prevent tipping. With the pendulum, ensure the clamp is tight and the bob cannot fly off. Always follow local health and safety guidelines for physics laboratories.
在进行旋转橡胶塞实验时，要佩戴护目镜以防绳子断裂。确保实验区域没有其他学生，不要以过高速度旋转橡胶塞。悬挂的槽码应连接牢固，并在下方放置柔软承接物（如泡沫垫）以防止重物掉落。对于弹簧振子，避免将弹簧拉伸超过弹性限度；使用带有沉重底座的支架以免倾倒。对于单摆实验，确保夹具拧紧，摆球不会飞出。务必始终遵守物理实验室的健康与安全指南。
10. Conclusion and Evaluation | 结论与评价
This experimental investigation provides strong supporting evidence for the theoretical models of circular and periodic motion. The linear relationships predicted—F ∝ ω² for fixed m and r, T² ∝ m_s for the spring oscillator, and T² ∝ L for the pendulum—are generally well confirmed, with small discrepancies attributable to friction, parallax errors, and timing inaccuracies. Through careful technique and multiple repeats, the derived values for the spring constant k and gravitational acceleration g can approach the true values within a few percent. The experience reinforces the importance of controlling variables, using graphical analysis, and understanding the limitations of laboratory apparatus. These experiments form a solid foundation for further study in rotational dynamics and wave phenomena.
本次实验探究为圆周运动和周期运动的理论模型提供了有力的支持性证据。所预测的线性关系——对于固定的 m 和 r 有 F ∝ ω²，对于弹簧振子有 T² ∝ m_s，对于单摆有 T² ∝ L——通常都能得到较好的验证，微小的偏差可归因于摩擦、视差误差和计时不准。通过仔细的操作和多次重复，推导出的弹簧劲度系数 k 和重力加速度 g 可以接近真实值，误差仅在百分之几以内。这一经历强化了控制变量、运用图像分析以及理解实验仪器局限性的重要性。这些实验为进一步学习转动动力学和波现象奠定了坚实的基础。
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