📚 IB Physics HL Pearson Textbook Experimental Investigations | IB物理HL实验探究：培生教材全解析
This article explores the core experimental investigations featured in the IB Physics HL Pearson textbook, emphasising practical skills, data analysis, and the handling of uncertainties. These investigations form the foundation for the internal assessment and deepen understanding of key concepts in mechanics, waves, electricity, and modern physics.
本文深入解析IB物理HL培生教材中的核心实验探究，重点关注实验技能、数据分析以及不确定度的处理。这些实验不仅是内部评估的基础，更能强化对力学、波动、电磁学和现代物理关键概念的理解。
1. Measuring g with a Simple Pendulum | 用单摆测量重力加速度
The simple pendulum offers a straightforward method to determine the acceleration due to gravity g by measuring the period T for oscillations at different lengths L. Small initial displacements (<10°) ensure the motion approximates simple harmonic.
单摆为测定重力加速度g提供了直接的方法，通过测量不同摆长L下的周期T实现。保持小角度摆动 (<10°) 可确保运动近似为简谐振动。
The period is given by T = 2π√(L/g). Squaring both sides gives T² = (4π²/g)L. Plotting T² against L yields a straight line through the origin, with slope = 4π²/g, from which g is calculated.
周期公式为 T = 2π√(L/g)，平方后得到 T² = (4π²/g)L。绘制 T² 对 L 的图像会得到一条过原点的直线，斜率等于 4π²/g，据此可计算出 g。
T² = (4π²/g) L
Major uncertainties stem from human reaction time when using a stopwatch (≈0.2 s) and the precise measurement of length to the centre of the bob. Timing 20 oscillations reduces the fractional uncertainty in T. The relative uncertainty in g is Δg/g = ΔL/L + 2(ΔT/T).
主要不确定度来源于使用秒表时的人为反应时间 (约0.2 s) 以及测量到摆球中心长度的精确性。计时20个周期可以减小周期T的相对不确定度。g的相对不确定度为 Δg/g = ΔL/L + 2(ΔT/T)。
| L (m) | Time for 20T (s) | T (s) | T² (s²) |
|---|---|---|---|
| 0.60 | 31.2 | 1.56 | 2.43 |
| 0.80 | 35.8 | 1.79 | 3.20 |
| 1.00 | 40.0 | 2.00 | 4.00 |
2. Young’s Modulus of a Wire | 金属丝的杨氏模量
Young’s modulus E characterises the elastic stiffness of a material. A long, thin wire is loaded with increasing masses, and the extension ΔL is measured using a vernier scale or travelling microscope. Stress = F/A and strain = ΔL/L₀.
杨氏模量E表征材料的弹性刚度。在长细金属丝上递增加载质量，用游标卡尺或移测显微镜测量伸长量ΔL。应力 = F/A，应变 = ΔL/L₀。
For a wire of diameter d, cross-sectional area A = πd²/4. The load F = mg. Plotting stress against strain produces a linear region whose gradient gives E.
对于直径为d的金属丝，截面积 A = πd²/4。负载 F = mg。绘制应力-应变图，其线性区域的斜率即为E。
E = (F/A) / (ΔL/L₀)
Key systematic errors include failing to straighten the wire initially (yielding a kink in the graph), parallax errors when reading the marker, and the weight of the hanger. Use a long wire (≈2 m) to magnify extension and reduce fractional uncertainty in ΔL.
主要的系统误差包括初始未拉直金属丝（导致图像出现曲折）、读数时的视差以及吊钩自身重量。使用长约2米的金属丝可以放大伸长量，减小ΔL的相对不确定度。
3. Ohm’s Law and Resistivity | 欧姆定律与电阻率
Ohm’s law states that the current I through a metallic conductor is proportional to the potential difference V at constant temperature. By varying V and recording I, a graph of V versus I yields resistance R as the gradient.
欧姆定律指出，在恒定温度下流过金属导体的电流I与电势差V成正比。改变V并记录I，绘制V-I图，斜率即为电阻R。
Resistivity ρ of the wire material is found from R = ρL/A. Measuring R for wires of the same material but different lengths L and plotting R against L gives a line with gradient ρ/A. Use a micrometer to measure diameter d at several points to find mean A.
材料的电阻率ρ可由 R = ρL/A 求得。测量同种材料不同长度L导线的电阻R，绘制R-L图，直线斜率为 ρ/A。用千分尺在多点测量直径d，取平均值计算截面积A。
Heating effects raise resistance and cause deviation from linearity; hence, keep currents low or switch off between readings. The uncertainty in ρ combines contributions from ΔR, ΔL, and Δd: Δρ/ρ = ΔR/R + ΔL/L + 2Δd/d.
热效应会使电阻升高，导致偏离线性；因此应使电流尽可能小或每次读数后断开电路。ρ的不确定度综合了ΔR、ΔL和Δd的影响：Δρ/ρ = ΔR/R + ΔL/L + 2Δd/d。
4. Refractive Index of a Glass Block | 玻璃砖的折射率
Using a ray box and a rectangular glass block, the angle of incidence i and angle of refraction r are measured with a protractor. Snell’s law gives the refractive index n = sin i / sin r.
利用光箱和矩形玻璃砖，用量角器测量入射角i和折射角r。根据斯涅耳定律，折射率 n = sin i / sin r。
Plotting sin i against sin r yields a straight line through the origin with gradient n. This graphical approach averages out random errors. Pins can be used to trace the ray path accurately; parallax must be avoided when aligning pins.
绘制 sin i 对 sin r 的图像，得到一条过原点的直线，斜率为n。这种图像法能平均随机误差。可以使用大头针精确描绘光线路径，对齐时必须消除视差。
Uncertainties arise from the thickness of the ray beam and the resolution of the protractor (±0.5°). To improve precision, measure angles for several values of i between 10° and 70°. Remember that the largest uncertainty in sin r occurs at small angles.
不确定度来源于光束的宽度和量角器的分辨力 (±0.5°)。为提高精确度，在10°至70°范围内取多个i值测量。注意 sin r 在小角度时不确定度最大。
5. Boyle’s Law Investigation | 玻意耳定律探究
Boyle’s law states that for a fixed mass of gas at constant temperature, pressure p is inversely proportional to volume V, so pV = constant. A syringe connected to a pressure sensor or Bourdon gauge allows data collection as volume is varied.
玻意耳定律指出，一定质量气体在温度不变时，压强p与体积V成反比，即 pV = 常数。将注射器连接压力传感器或波登气压计，改变体积并采集数据。
Plotting p against 1/V should give a straight line passing through the origin, though in practice a small offset may appear due to the dead volume of tubing. Alternatively, a graph of log p versus log V yields a gradient of −1.
绘制 p 对 1/V 的图，应得到一条过原点的直线，但实际中由于管道死体积的存在可能会出现微小截距。也可绘制 log p 对 log V 的图，斜率应为−1。
Slowly compress the gas to maintain thermal equilibrium with surroundings. Major uncertainties include reading the volume scale and possible leaks. Use thin-walled tubing to minimise dead volume and repeat measurements at constant temperature.
缓慢压缩气体以保持与周围环境的热平衡。主要不确定度包括体积刻度的读数和可能的气体泄漏。使用薄壁软管以减少死体积，并在恒温下重复测量。
6. Wavelength of Light by Double‑Slit Interference | 双缝干涉测光波波长
Young’s double‑slit experiment demonstrates the wave nature of light. Monochromatic light passes through two narrow slits separated by distance d, producing a fringe pattern on a screen distance D away. The fringe spacing Δx is given by Δx = λD/d.
杨氏双缝实验证实了光的波动性。单色光通过相距为d的两条狭缝，在距离D处的屏幕上产生干涉条纹。条纹间距 Δx 由 Δx = λD/d 给出。
Measure the distance across several bright fringes (e.g. 10) to determine Δx more accurately. Use a travelling microscope or a ruler with a magnifier. The slit separation d is often provided; otherwise, it can be measured using a laser and diffraction method.
测量多个亮条纹（如10条）间的距离以更精确地确定Δx。可使用移测显微镜或带放大镜的直尺。狭缝间距d通常给定，否则可通过激光衍射法测量。
Ensure the screen is perpendicular to the optical bench. The biggest uncertainty is usually in Δx, especially if fringes are wide and diffuse. Reducing ambient light improves contrast. The wavelength uncertainty Δλ/λ = Δ(Δx)/Δx + ΔD/D + Δd/d.
确保屏幕垂直于光具座。最大的不确定度通常来自Δx，尤其是当条纹宽且模糊时。降低环境光可提高对比度。波长不确定度 Δλ/λ = Δ(Δx)/Δx + ΔD/D + Δd/d。
7. Magnetic Field of a Solenoid | 螺线管的磁场
The magnetic flux density B inside a long solenoid is given by B = μ₀nI, where n is the number of turns per unit length and I the current. A Hall probe placed along the axis maps the field profile.
长螺线管内部的磁通密度为 B = μ₀nI，其中n为单位长度匝数，I为电流。将霍尔探头沿轴线放置，可以描绘出磁场分布。
Vary the current and plot B versus I to verify the linear relationship and determine μ₀ from the gradient, as n is known. The Earth’s magnetic field must be subtracted by zeroing the probe before applying current.
改变电流并绘制 B 对 I 图像，可验证线性关系并从斜率确定 μ₀ (已知n)。在通电前必须对探头进行调零，以减去地磁场的影响。
Keep currents moderate to avoid heating the coil. The Hall probe should be calibrated and oriented perpendicular to the field. Systematic uncertainty includes probe alignment and the non‑ideal uniformity near the solenoid ends; measurements at the centre minimise this.
保持中等电流以避免线圈发热。霍尔探头需经过校准并垂直于磁场放置。系统不确定度包括探头对准以及螺线管两端附近的非理想均匀性；在中心进行测量可将此影响降至最低。
8. Simulation of Radioactive Decay | 放射性衰变的模拟
Radioactive decay follows the exponential law N = N₀ e^(−λt). A common simulation uses a large number of dice or coins. Each throw represents a fixed time interval; dice showing a specific face (e.g. ‘6’) are deemed to have decayed and are removed.
放射性衰变遵循指数规律 N = N₀ e^(−λt)。常见的模拟实验使用大量骰子或硬币。每次抛掷代表一个固定的时间间隔；出现特定一面（如“6”）的骰子被视为已衰变并移除。
The number of remaining dice N is recorded after each throw. Plotting ln N against throw number yields a straight line of slope −λ’ (decay probability per throw). Half‑life T½ can be determined from the graph or the decay constant.
记录每次抛掷后剩余的骰子数N。绘制 ln N 对抛掷次数的图，得到斜率为−λ’ (每次抛掷的衰变概率) 的直线。半衰期 T½ 可从图像或衰变常数得出。
This analogue illustrates the random nature of decay and the statistical fluctuations observed in small samples. Averaging multiple runs smooths out fluctuations and improves the estimate of T½. Real experiments can use a Geiger‑Müller tube and a short‑lived isotope like protactinium‑234.
该模拟说明了衰变的随机性以及小样本中观察到的统计涨落。多次运行取平均可以平滑涨落，提高半衰期的估计精度。真实实验可使用盖革‑缪勒计数管和短寿命同位素，如镤‑234。
9. Internal Resistance and EMF of a Cell | 电池的内阻与电动势
A real cell can be modelled as an emf ε in series with an internal resistance r. The terminal voltage V across the cell when current I flows is V = ε − I r. By varying an external load resistor R, pairs of V and I are recorded.
一个真实的电池可以等效为一个电动势ε与内阻r串联。当流过电流I时，电池端电压 V = ε − I r。通过改变外部负载电阻R，记录多组V和I值。
Plot V on the y‑axis and I on the x‑axis. The intercept on the voltage axis gives ε, and the slope gives −r. Alternatively, use a graph of R vs 1/I if a variable resistor box is used.
将V置于纵轴、I置于横轴绘图。电压轴上的截距即为ε，斜率的绝对值为r。若使用电阻箱，还可绘制 R 对 1/I 的图像。
High currents cause internal heating, altering r; take readings quickly and allow the cell to rest. The uncertainty in ε and r can be obtained from the intercept and slope of error lines. Use a digital voltmeter and ammeter with appropriate ranges to minimise loading errors.
大电流会导致内阻发热改变r值；需快速读数并让电池间歇性恢复。ε和r的不确定度可通过最佳拟合线与误差线的截距和斜率求得。使用适当量程的数字电压表和电流表以减少负载误差。
10. Specific Heat Capacity by Electrical Heating | 电加热法测比热容
The specific heat capacity c of a substance is the energy needed to raise the temperature of 1 kg by 1 K. A metal block (e.g. aluminium) is heated with an immersion heater of known power P for time t. The electrical energy supplied is P t = V I t.
比热容c是使1 kg物质温度升高1 K所需的热量。用已知功率P的电热器加热金属块（如铝）时间为t，提供的电能为 P t = V I t。
Assuming no heat losses, the energy gained by the block is m c Δθ, where Δθ is the temperature rise. Thus c = V I t / (m Δθ). Monitoring temperature continuously with a data logger gives a reliable Δθ.
假设无热损失，金属块获得的热量为 m c Δθ，其中Δθ为温升。因此 c = V I t / (m Δθ)。使用数据采集器连续监测温度可得到可靠的Δθ。
Heat losses to the surroundings are the dominant error. Insulating the block with lagging and starting a few degrees below room temperature so that the average temperature equals ambient helps compensate. The uncertainty in c is dominated by Δθ measurement; using a sensitive thermometer improves precision.
向环境散失的热量是主要的误差来源。用保温材料包裹金属块，并使初始温度略低于室温，使得平均温度等于环境温度，有助于补偿热损失。c的不确定度主要由Δθ的测量决定，使用高灵敏度温度计可提高精度。
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