📚 Tools for Physics | 物理学的工具箱
Physics is built on a foundation of precise measurement, robust mathematics, and careful data analysis. To succeed in IB Physics, you must become fluent in the essential tools of the trade — from SI units and scientific notation to vector manipulation and uncertainty propagation. Mastering these tools will allow you not only to perform experiments but also to interpret physical phenomena with confidence and clarity.
物理学建立在精密测量、严格数学和细致数据分析的基础之上。要在 IB 物理中取得好成绩，你必须熟练掌握这门学科的基本工具——从国际单位制和科学记数法，到向量操作和不确定度传播。掌握这些工具不仅能让你顺利完成实验，还能让你自信、清晰地解释物理现象。
1. Measurement and SI Units | 测量与国际单位制
Every physical quantity is expressed as a numerical value and a unit. The International System of Units (SI) defines seven base quantities, each with a base unit. All other units are derived from these base units through multiplication and division.
每一个物理量都表示为数值和单位的组合。国际单位制 (SI) 定义了七个基本量，每个基本量都有一个基本单位。其他所有单位都是由这些基本单位通过乘除导出的。
| Base Quantity | Base Unit | Symbol |
|---|---|---|
| Mass | kilogram | kg |
| Length | metre | m |
| Time | second | s |
| Electric current | ampere | A |
| Temperature | kelvin | K |
| Amount of substance | mole | mol |
| Luminous intensity | candela | cd |
牢记这七个基本量和单位是处理量纲分析、单位换算和导出单位的基础。
In IB Physics, you will frequently encounter derived units such as the newton (N), which is equivalent to kg m s⁻², and the joule (J), which is kg m² s⁻². Being able to express derived units in base units is a crucial skill for checking the consistency of equations.
在 IB 物理中，你会经常遇到牛顿 (N) 等导出单位，N 相当于 kg m s⁻²，焦耳 (J) 相当于 kg m² s⁻²。能够用基本单位表示导出单位是检验方程是否一致的关键技能。
2. Scientific Notation and Orders of Magnitude | 科学记数法与数量级
nteger. This is essential when dealing with physical constants and astronomical distances.
科学记数法使我们能够紧凑地表示非常大或非常小的数字。数字写作 a × 10ⁿ 的形式，其中 1 ≤ a < 10，n 为整数。在处理物理常数和天文距离时，这是必不可少的。
The order of magnitude of a number is the power of ten when the number is expressed in scientific notation, often rounded to the nearest integer. It is a quick way to compare the scale of different quantities.
一个数的数量级是该数以科学记数法表示时 10 的幂次，通常四舍五入到最近的整数。它是快速比较不同量级的一种方法。
| Quantity | Approximate Value | Order of Magnitude |
|---|---|---|
| Radius of a proton | 1 × 10⁻¹⁵ m | 10⁻¹⁵ |
| Radius of a hydrogen atom | 5 × 10⁻¹¹ m | 10⁻¹⁰ |
| Height of a human | 1.7 × 10⁰ m | 10⁰ |
| Radius of the Earth | 6.4 × 10⁶ m | 10⁷ |
| Distance to the nearest star | 4 × 10¹⁶ m | 10¹⁶ |
Familiarity with these orders of magnitude helps you develop physical intuition and is often tested in IB exam questions asking you to estimate quantities.
熟悉这些数量级有助于培养物理直觉，在 IB 考试中经常会要求你对某些量进行估算。
3. Significant Figures | 有效数字
The number of significant figures (s.f.) in a measurement reflects its precision. Rules for identifying s.f. are: all non-zero digits are significant; zeros between non-zero digits are significant; leading zeros are not; trailing zeros after a decimal point are significant.
测量值中有效数字的位数反映了其精度。识别有效数字的规则是：所有非零数字都是有效数字；非零数字之间的零是有效数字；前导零不是；小数点后的尾随零是有效数字。
When multiplying or dividing, the result should be expressed to the same number of significant figures as the least precise measurement used. When adding or subtracting, the result should have the same number of decimal places as the measurement with the fewest decimal places.
当进行乘除运算时，结果的有效数字位数应与所用测量值中精度最低的保持一致。当进行加减运算时，结果的小数位数应与小数位数最少的测量值相同。
Example: 12.3 cm (3 s.f.) × 2.5 cm (2 s.f.) = 30.75 cm² → rounded to 31 cm² (2 s.f.).
示例：12.3 cm（3 位有效数字） × 2.5 cm（2 位有效数字） = 30.75 cm² → 四舍五入为 31 cm²（2 位有效数字）。
4. Errors and Uncertainties | 误差与不确定性
st scale division, while digital instruments carry an uncertainty of ± the smallest digit.
没有测量是绝对精确的。测量不确定度表示真值可能存在的范围。模拟仪器的不确定度通常为 ± 最小刻度的一半，而数字仪器的不确定度为 ± 最小位数。
Random errors cause measurements to be scattered around the true value and can be reduced by taking repeat readings and calculating the mean. Systematic errors shift all readings in one direction and are not eliminated by averaging; they require instrument calibration or correction.
随机误差导致测量值分散在真值周围，可以通过多次重复测量并计算平均值来减小。系统误差使所有读数向同一方向偏移，取平均值无法消除；需要校准仪器或进行修正。
5. Absolute, Relative and Percentage Uncertainties | 绝对、相对与百分数不确定度
measurement. For a measured length of 2.50 ± 0.05 cm, the absolute uncertainty is 0.05 cm.
绝对不确定度 (Δx) 用与测量值相同的单位表示。例如，测量长度为 2.50 ± 0.05 cm，绝对不确定度为 0.05 cm。
Relative uncertainty = Δx / x (no units). Percentage uncertainty = (Δx / x) × 100% uncertainties of different measurements or when propagating uncertainties in calculations.
相对不确定度 = Δx / x（无单位）。百分数不确定度 = (Δx / x) × 100%。这些形式在比较不同测量值的不确定度或在计算中传播不确定度时特别有用。
6. Propagation of Uncertainties | 不确定度的传播
When measurements are combined, their uncertainties also combine. The rules depend on the mathematical operation.
当测量值组合在一起时，它们的不确定度也会组合。规则取决于数学运算。
- Addition / Subtraction:
If y = a ± b, then Δy = Δa + Δb
- Multiplication / Division:
If y = a × b or y = a ÷ b, then Δy/y = Δa/a + Δb/b
- Power rule:
If y = aⁿ, then Δy/y = |n| Δa/a
For more complex functions, IB students are not required to use calculus-based propagation, but you should be able to substitute extreme values (max/min method) to estimate the uncertainty range.
对于更复杂的函数，IB 不要求学生使用基于微积分的传播方法，但应能通过代入极值（最大/最小法）来估计不确定度范围。
us ΔA = 0.0583 × 6.00 ≈ 0.35 m². Area = 6.0 ± 0.4 m² (expressed to appropriate precision).
示例：若长度 L = (2.00 ± 0.05) m，宽度 W = (3.00 ± 0.10) m，则面积 A = L×W = 6.00 m²。相对不确定度：ΔL/L = 0.025，ΔW/W = 0.0333，所以 ΔA/A = 0.0583，因此 ΔA = 0.0583 × 6.00 ≈ 0.35 m²。面积 = 6.0 ± 0.4 m²（精确到适当精度）。
7. Vectors and Scalars | 向量与标量
A scalar quantity has only magnitude, while a vector quantity has both magnitude and direction. Recognising the difference is fundamental for solving mechanics problems correctly.
标量只有大小，而向量既有大小又有方向。区分两者是正确解决力学问题的基础。
| Scalar | Vector |
|---|---|
| Distance | Displacement |
| Speed | Velocity |
| Mass | Weight |
| Energy | Momentum |
| Temperature | Acceleration |
Vectors are typically represented by bold symbols or arrows, e.g., F or v⃗. In IB Physics, you will work with vectors in free-body diagrams, kinematics, and fields.
向量通常用粗体符号或箭头表示，例如 F 或 v⃗。在 IB 物理中，你将在自由体图、运动学和场中用到向量。
8. Vector Components and Addition | 向量分解与加法
To add vectors, we can use the parallelogram method or break them into perpendicular components (usually horizontal and vertical, or along x and y axes).
要相加向量，我们可以使用平行四边形法，或将其分解为相互垂直的分量（通常是水平和垂直分量，或沿 x 轴和 y 轴）。
For a vector of magnitude R at an angle θ to the horizontal:
x-component = R cos θ ; y-component = R sin θ
对于大小为 R、与水平方向夹角为 θ 的向量：
x 分量 = R cos θ ; y 分量 = R sin θ
When adding two vectors, simply add their respective components. The magnitude of the resultant can then be found using Pythagoras' theorem, and the direction using tan⁻¹(y/x). This technique is essential for handling forces in equilibrium, projectile motion, and electromagnetic fields.
当相加两个向量时，只需将各自的分量相加。合成向量的大小可以用勾股定理求得，方向则用 tan⁻¹(y/x) 求出。这种技巧在处理平衡力、抛体运动和电磁场问题时至关重要。
9. Graphical Analysis and Linearization | 图形分析与线性化
Plotting experimental data on a graph is one of the most powerful tools in physics. A straight-line graph allows you to determine relationships between variables, calculate gradients, and find intercepts.
将实验数据绘制在图上，是物理学中最强大的工具之一。直线图可以帮助你确定变量之间的关系，计算斜率和截距。
If the raw data produces a curve, you can often linearize it by plotting transformed variables. For example, if theory predicts y is proportional to x², a graph of y vs x² will yield a straight line through the origin. The slope of that line then equals the proportionality constant.
如果原始数据产生曲线，通常可以通过绘制变换后的变量来实现线性化。例如，如果理论预测 y 与 x² 成正比，那么绘制 y 对 x² 的图将得到一条过原点的直线。该直线的斜率就等于比例常数。
IB Physics labs frequently require you to linearize data, use error bars, and calculate the uncertainty in the slope by drawing maximum and minimum gradient lines.
IB 物理实验经常要求你对数据进行线性化，使用误差棒，并通过绘制最大和最小斜率线来计算斜率的不确定度。
10. Trigonometric Tools | 三角函数工具
Trigonometry is indispensable in physics, especially when resolving vectors, analyzing waves, and studying oscillations. Key relations in a right-angled triangle:
三角学在物理学中不可或缺，尤其是在分解向量、分析波动和研究振动时。直角三角形中的关键关系：
in θ = opposite / hypotenuse ; cos θ = adjacent / hypotenuse ; tan θ = opposite / adjacent
sin θ = 对边 / 斜边 ; cos θ = 邻边 / 斜边 ; tan θ = 对边 / 邻边
extremely useful in topics like simple harmonic motion and single-slit diffraction. Note that IB Physics expects you to work with radians when using these approximations.
对于小角度（θ 以弧度为单位），近似公式 sin θ ≈ θ 和 tan θ ≈ θ 在简谐运动、单缝衍射等课题中非常有用。注意，IB 物理要求你在使用这些近似时必须用弧度。
Familiarity with sine and cosine functions is also central to understanding alternating current, wave phase differences, and circular motion.
熟练掌握正弦和余弦函数，对于理解交流电、波的相位差和圆周运动也是至关重要的。
11. Using Calculus in Physics | 物理中的微积分初探
Although IB Physics is not a calculus-intensive course, the language of differential and integral calculus appears in several areas. Instantaneous velocity is the derivative of displacement with respect to time, and acceleration is the derivative of velocity:
虽然 IB 物理并非一门微积分密集型的课程，但微分和积分的语言出现在多个领域。瞬时速度是位移对时间的导数，加速度是速度对时间的导数：
v = dx/dt ; a = dv/dt
Conversely, displacement can be found as the area under a velocity–time graph, mirroring integration. In the HL topics, you will use the idea that work done by a variable force is the area under a force–displacement graph.
反过来，位移可以通过速度-时间图下的面积求得，这对应积分。在 HL 课题中，你会用到变力做功是力-位移图下面积的概念。
Understanding these basic calculus relationships gives you deeper insight into the unified nature of physics and prepares you for university-level study.
理解这些基本的微积分关系，可以让你更深入地洞察物理的统一性，并为大学阶段的学习做好准备。
12. Error Bars and Best-Fit Lines | 误差棒与最佳拟合线
When plotting data points on a graph, each point should have error bars that reflect its uncertainty in the x and y directions (if both are known). The best-fit line is then drawn through the data points, ideally passing through the error bars.
在图上绘制数据点时，每个点都应有误差棒，以反映其在 x 和 y 方向上的不确定度（如果两者都已知）。然后绘制最佳拟合线，使其穿过数据点，理想情况下应通过误差棒。
To estimate the uncertainty in the slope, draw two additional lines: the steepest and the shallowest lines that still reasonably pass through the error bars. The uncertainty is then:
要估算斜率的不确定度，需另外绘制两条线：仍能合理穿过误差棒的最陡线和最浅线。然后不确定度为：
Δslope = (slope_max − slope_min) / 2
This technique provides a realistic assessment of the error in derived quantities and is a standard requirement in IB internal assessments.
这种技术提供了对导出量误差的切合实际的评估，是 IB 内部评估中的标准要求。
Remember that a line should not be forced through the origin unless there is a theoretical justification. The intercept on a graph can reveal systematic errors in your experiment.
请记住，除非有理论依据，否则不应强迫直线经过原点。图上的截距可以揭示实验中的系统误差。
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