5.5 & 5.6: Determining Density – Methods and Applications | 5.5与5.6：密度的测定——方法与运用

📚 5.5 & 5.6: Determining Density – Methods and Applications | 5.5与5.6：密度的测定——方法与运用

In the Edexcel IGCSE Physics specification, topics 5.5 and 5.6 focus on practical methods for determining the density of regular solids, irregular solids, and liquids. Density is a fundamental physical property that links mass and volume, and it has countless real-world applications, from predicting whether an object floats to designing ships and aircraft. This article provides a clear, exam-focused guide to these syllabus points, including step-by-step experimental procedures, worked examples, and tips for avoiding common errors.

在 Edexcel IGCSE 物理课程大纲中，知识点 5.5 和 5.6 聚焦于测定规则固体、不规则固体和液体密度的实验方法。密度是联系质量与体积的基本物理属性，其应用极为广泛——从判断物体是否漂浮到设计船舶与飞机。本文针对这些考点提供清晰、以考试为导向的讲解，包括分步实验步骤、例题以及常见错误的规避技巧。

1. What is Density? | 什么是密度？

Density is defined as the mass of an object per unit volume. It tells us how tightly packed matter is within a given space. A material with a high density, such as iron, has a large amount of mass in a small volume, whereas a low-density material, such as wood, has less mass in the same volume. In the IGCSE syllabus, the concept of density is essential for understanding floatation, pressure in fluids, and material properties.

密度定义为单位体积内的质量。它反映物质在空间中的紧密程度。高密度材料如铁，在较小体积中含有较大质量；而低密度材料如木材，在相同体积中质量较小。在 IGCSE 课程中，密度概念是理解漂浮、流体压强和材料性质的关键。

2. The Density Formula | 密度公式

The mathematical relationship between density, mass and volume is expressed using the symbol ρ (rho). The formula is:

ρ = m / V

Where ρ is density, m is mass, and V is volume. The SI unit of density is kilograms per cubic metre (kg/m³). However, in many school experiments, grams per cubic centimetre (g/cm³) is also used. Remember that 1 g/cm³ is equal to 1000 kg/m³. In IGCSE problems, you must be able to rearrange the formula to find mass or volume when the other two quantities are known.

密度、质量与体积之间的数学关系用符号 ρ（rho）表示，公式为：

ρ = m / V

其中 ρ 表示密度，m 表示质量，V 表示体积。密度的国际单位是千克每立方米（kg/m³）。但在许多学校实验中，也常用克每立方厘米（g/cm³）。请记住，1 g/cm³ 等于 1000 kg/m³。在 IGCSE 试题中，你需要能够对公式进行变形，在已知其中两个量时求出质量或体积。

3. Measuring Density of a Regular Solid | 测量规则固体的密度

For a regular solid, such as a rectangular block or a cylinder, the volume can be calculated directly from its dimensions using simple geometric formulas. The method involves three main steps. First, measure the mass of the solid using a digital balance or a pan balance. Second, measure the necessary dimensions with a ruler or vernier callipers. For a cuboid, measure its length, width and height; for a cylinder, measure its diameter and height. Third, calculate the volume using the appropriate formula, then substitute the mass and volume into the density equation.

对于规则固体，如长方体或圆柱体，可以直接利用几何公式根据尺寸计算体积。该方法主要包含三个步骤：第一步，使用电子天平或托盘天平测量固体质量；第二步，用刻度尺或游标卡尺测量所需尺寸——对于长方体需测量长、宽、高，对于圆柱体需测量直径和高；第三步，用相应公式计算体积，然后将质量和体积代入密度公式。

For a cuboid, the volume is length × width × height (l × w × h). For a cylinder, the volume is πr²h, where r is the radius and h is the height. Once the volume is known, divide the mass by the volume to obtain the density. Always record the measurements with appropriate precision, and repeat the experiment to get an average.

长方体的体积为长 × 宽 × 高（l × w × h）。圆柱体的体积为 πr²h，其中 r 是半径，h 是高。得到体积后，用质量除以体积即得密度。测量时要确保适当的精度，并重复实验以取平均值。

4. Measuring Density of an Irregular Solid | 测量不规则固体的密度

Irregular solids, such as a small stone or a metallic object, have shapes that cannot be easily measured with a ruler. To find their volume, we use the principle of water displacement. First, measure the mass of the irregular solid using a balance. Then, fill a measuring cylinder (also called a graduated cylinder) with a known volume of water. Record the initial volume. Gently lower the solid into the water so it is fully submerged, and record the new volume. The volume of the solid is equal to the difference between the final and initial water levels.

不规则固体，如小石块或金属物体，其形状无法直接用刻度尺测量。为了求其体积，我们使用排水法。首先，用天平测量不规则固体的质量；然后，在一个量筒中装入已知体积的水，记录初始水位；接着，将固体轻轻完全浸入水中，记录新的水位；固体的体积等于最终水位与初始水位之差。

The density is then calculated as mass divided by the displaced volume. This method works because the solid displaces exactly its own volume of water when fully immersed. It is important to ensure that the solid does not float and that no water splashes out of the measuring cylinder. For small floating objects, a sinker may be attached to force the object underwater.

密度则等于质量除以被排开水的体积。该方法可行是因为固体完全浸没时排开水的体积恰好等于其自身体积。需要确保固体不漂浮，且水不会溅出量筒。对于会漂浮的小物体，可以加一个重物（坠子）将其压入水中。

5. Measuring Density of a Liquid | 测量液体的密度

The density of a liquid can be determined directly using a measuring cylinder and a balance. First, measure the mass of an empty measuring cylinder. Then, pour a known volume of the liquid into the cylinder and measure the combined mass. Subtract the mass of the empty cylinder from the total mass to obtain the mass of the liquid. Finally, divide the mass of the liquid by its volume to find the density.

液体的密度可以直接用量筒和天平测量。首先，测量空量筒的质量；然后，将一定体积的液体倒入量筒，测量总质量；用总质量减去空量筒质量，得到液体的质量；最后，用液体质量除以体积，即得密度。

An alternative, more accurate method involves using a density bottle (or specific gravity bottle). A density bottle has a precisely known volume. You weigh the bottle empty, then filled with the liquid, and calculate the density using the same principle. This reduces errors caused by parallax when reading the meniscus in a measuring cylinder.

另一种更精确的方法是使用密度瓶（比重瓶）。密度瓶具有精确已知的体积。先称空瓶质量，再称装满液体后的质量，按照同样原理计算密度。这种方法可减少用量筒读数时因凹液面视差产生的误差。

6. Worked Example | 例题演示

Example 1: A cuboid of metal has length 4.0 cm, width 2.0 cm and height 1.5 cm. Its mass is 96 g. Calculate the density of the metal in g/cm³.

例题1：一块金属长方体长 4.0 cm，宽 2.0 cm，高 1.5 cm，质量为 96 g。计算该金属的密度（单位 g/cm³）。

Solution: Volume = l × w × h = 4.0 × 2.0 × 1.5 = 12 cm³. Density = mass / volume = 96 / 12 = 8.0 g/cm³. This is consistent with a material such as brass or copper.

解答：体积 = l × w × h = 4.0 × 2.0 × 1.5 = 12 cm³。密度 = 质量 / 体积 = 96 / 12 = 8.0 g/cm³。这与黄铜或铜等材料的数据相符。

Example 2: A stone has a mass of 50 g. When placed into a measuring cylinder containing 30 cm³ of water, the water level rises to 48 cm³. Find the density of the stone.

例题2：一块石头质量为 50 g。将它放入盛有 30 cm³ 水的量筒中，水位上升到 48 cm³。求石头的密度。

Solution: Volume of stone = 48 − 30 = 18 cm³. Density = 50 / 18 = 2.78 g/cm³ (to two decimal places).

解答：石头体积 = 48 − 30 = 18 cm³。密度 = 50 / 18 = 2.78 g/cm³（保留两位小数）。

7. Comparing Methods | 方法比较

Type of Substance	Volume Measurement	Equipment Used	Advantages	Disadvantages
Regular solid	Geometric formula based on dimensions	Ruler, vernier callipers, balance	Simple; no liquid needed; high accuracy if dimensions are exact	Only works for regular shapes; measurement of small dimensions can be difficult
Irregular solid	Water displacement	Measuring cylinder, water, string	Works for any solid; quick	Solid must not react with water; may be inaccurate if solid floats; meniscus reading errors
Liquid	Known volume from measuring cylinder or density bottle	Measuring cylinder, density bottle, balance	Direct and straightforward; density bottle increases precision	Evaporation losses; might need temperature control; cleaning required between samples

The table above summarises the three typical methods and their strengths and limitations. For exam questions, you must be able to describe the procedure and explain why one method may be preferable in a given scenario.

上表总结了三种常见方法及其优缺点。在考试中，你需要能够描述实验步骤，并解释在特定情境下为什么某种方法更可取。

8. Units and Conversion | 单位与换算

In calculations, it is essential to use consistent units. The SI unit of density is kg/m³, but school experiments often produce results in g/cm³. To convert from g/cm³ to kg/m³, multiply by 1000. For example, water has a density of 1.0 g/cm³, which is equal to 1000 kg/m³. Conversely, to convert kg/m³ to g/cm³, divide by 1000.

在计算中，使用一致的单位至关重要。密度的国际单位是 kg/m³，但学校实验常得到 g/cm³ 的结果。将 g/cm³ 转换为 kg/m³ 时，乘以 1000。例如，水的密度为 1.0 g/cm³，等于 1000 kg/m³。反之，将 kg/m³ 转换为 g/cm³ 时，除以 1000。

When using the density formula, if mass is in grams and volume is in cubic centimetres, the density will be in g/cm³. If mass is in kilograms and volume is in cubic metres, the density will be in kg/m³. Always check your units before substituting values.

使用密度公式时，若质量以克为单位、体积以立方厘米为单位，则密度单位为 g/cm³。若质量以千克为单位、体积以立方米为单位，则密度单位为 kg/m³。代入数值前务必检查单位。

9. Common Errors and Exam Tips | 常见错误与考试提示

Error: Misreading the meniscus. When reading water level in a measuring cylinder, always read at eye level at the bottom of the meniscus. 错误：读数时视线偏高。读取量筒水位时，要平视凹液面底部。
Error: Forgetting to subtract the mass of the container. In liquid experiments, the balance reading includes the container. Subtract the container’s mass to get the liquid mass. 错误：忘记减去容器质量。在液体实验中，天平读数包括容器质量，需减去容器质量以获得液体质量。
Error: Using the wrong units. Forgetting to convert cm³ to m³ when required. Keep the same unit system throughout the calculation. 错误：单位使用错误。需要换算时忘记将 cm³ 转换为 m³。整个计算过程中要保持单位系统一致。
Tip: Repeat and average. Take multiple measurements to reduce random errors and improve reliability. 提示：重复测量并取多次结果的平均值，可减少偶然误差，提高可靠性。
Tip: Describe the method clearly. In exam questions, include full details of the equipment, steps and what you would measure. 提示：在实验题中，要清晰描述所需器材、步骤以及要测量什么量。

10. Conclusion | 结论

In conclusion, determining the density of solids and liquids is a core practical skill in Edexcel IGCSE Physics. For a regular solid, measure the dimensions to calculate volume; for an irregular solid, use water displacement; for a liquid, measure a known volume and calculate its mass. All methods ultimately rely on the equation ρ = m/V. Mastery of these techniques, including correct unit handling, accurate readings, and clear experimental descriptions, will help you succeed in your IGCSE examination.

总之，测定固体和液体的密度是 Edexcel IGCSE 物理中的核心实验技能。对于规则固体，测量尺寸计算体积；对于不规则固体，使用排水法；对于液体，测量一定体积的质量。所有方法最终都依赖 ρ = m/V 这一公式。掌握这些技巧，包括正确处理单位、准确读数以及清晰描述实验，将帮助你在 IGCSE 考试中取得成功。

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5.5 & 5.6: Determining Density – Methods and Applications | 5.5与5.6：密度的测定——方法与运用