Density & Pressure | 密度与压强

📚 Density & Pressure | 密度与压强

This article explores two closely related topics in IGCSE Physics: density and pressure. We will examine their definitions, formulas, practical measurement methods and real-world applications, with a special focus on the Edexcel specification.

本文深入探讨 IGCSE 物理中两个密切相关的话题：密度与压强。我们将学习它们的定义、公式、实际测量方法以及现实世界中的应用，并特别关注 Edexcel 考试大纲的要求。

1. Density: Definition and Formula | 密度：定义与公式

Density is defined as the mass of a substance per unit volume. It tells us how much matter is packed into a given space. The quantity symbol for density is the Greek letter rho (ρ).

密度定义为物质单位体积的质量。它告诉我们一定空间内含有多少物质。密度的符号是希腊字母 ρ（rho）。

The formula for density is:

ρ = m / V

where ρ is density, m is mass and V is volume. In one example, a metal block has a mass of 2 kg and a volume of 0.001 m³. Its density is therefore 2 / 0.001 = 2000 kg/m³.

其中 ρ 是密度，m 是质量，V 是体积。例如，一个金属块的质量是 2 kg，体积是 0.001 m³，所以它的密度为 2 / 0.001 = 2000 kg/m³。

Density is an important property of materials. If two objects have the same volume, the one with the greater mass has the greater density. Similarly, if two objects have the same mass, the one with the smaller volume has the greater density.

密度是材料的重要属性。如果两个物体体积相同，质量较大的那个密度更大；如果两个物体质量相同，体积较小的那个密度更大。

2. Units and Typical Densities | 单位与常见密度

In SI units, mass is measured in kilograms (kg), volume in cubic metres (m³), and density in kilograms per cubic metre (kg/m³). In the laboratory, density is often expressed in grams per cubic centimetre (g/cm³).

在国际单位制中，质量用千克（kg）测量，体积用立方米（m³）测量，密度用千克每立方米（kg/m³）表示。在实验室中，密度常用克每立方厘米（g/cm³）表示。

It is essential to recall the conversion: 1 g/cm³ = 1000 kg/m³. For example, water has a density of 1 g/cm³, which equals 1000 kg/m³.

必须牢记换算关系：1 g/cm³ = 1000 kg/m³。例如，水的密度为 1 g/cm³，即 1000 kg/m³。

The table below lists the densities of several common substances.

下表列出了几种常见物质的密度。

Substance \| 物质	Density / kg m⁻³ \| 密度 / kg m⁻³
Water \| 水	1000
Ice \| 冰	920
Aluminium \| 铝	2700
Iron \| 铁	7870
Air \| 空气	1.2

Another useful concept is relative density, also called specific gravity. It is the ratio of the density of a substance to the density of water. Since it is a ratio, it has no units.

另一个有用概念是相对密度，也称比重。它是物质的密度与水密度的比值。由于这是一个比值，因此没有单位。

3. Measuring the Density of Regular Solids | 测量规则固体的密度

For a regularly shaped solid such as a cuboid, cylinder or sphere, the density can be found by measuring its mass and calculating its volume from its dimensions.

对于长方体、圆柱体或球体等形状规则的固体，可以通过测量质量并根据外形尺寸计算体积来求密度。

Measure the mass of the solid using a digital balance.

用电子天平测量固体的质量。
Use a ruler or vernier callipers to measure the necessary dimensions: for a cuboid, measure length, width and height; for a cylinder, measure diameter and height.

用刻度尺或游标卡尺测量必要的尺寸：长方体需要长、宽、高；圆柱体需要直径和高。
Calculate the volume. For a cuboid, V = length × width × height. For a cylinder, V = πr²h, where r is radius and h is height.

计算体积。长方体 V = 长 × 宽 × 高；圆柱体 V = πr²h，其中 r 是半径，h 是高。
Substitute the measured mass and volume into ρ = m/V.

将测得的质量和体积代入 ρ = m/V。

For example, a cuboid has dimensions 2 cm × 3 cm × 5 cm. Its volume is 30 cm³. If its mass is 240 g, its density is 240 / 30 = 8 g/cm³, which is equal to 8000 kg/m³.

例如，一个长方体的尺寸为 2 cm × 3 cm × 5 cm，体积为 30 cm³。若其质量为 240 g，则密度为 240 / 30 = 8 g/cm³，等于 8000 kg/m³。

4. Measuring the Density of Irregular Solids and Liquids | 测量不规则固体和液体的密度

Irregular solids have no simple formula for volume, so we use the displacement method with a measuring cylinder.

不规则固体没有简单的体积公式，因此我们使用量筒的排水法来测定体积。

Measure the mass of the irregular solid using a balance.

用天平测量不规则固体的质量。
Pour a known volume of water, V₁, into a measuring cylinder.

向量筒中倒入已知体积的水，记为 V₁。
Carefully lower the object into the water. Record the new volume V₂.

小心地将物体浸入水中，记录新的体积 V₂。
The volume of the object is V₂ – V₁. Then density = m / (V₂ – V₁).

物体的体积为 V₂ – V₁。然后密度 = m / (V₂ – V₁)。

For a liquid, such as cooking oil, we can find its density using an empty beaker and a measuring cylinder.

对于液体（如食用油），可以使用空烧杯和量筒来测定其密度。

Measure the mass of an empty beaker, m₁.

测量空烧杯的质量，记为 m₁。
Pour a known volume V of liquid into the measuring cylinder.

向量筒中倒入一定体积 V 的液体。
Pour the liquid into the beaker and measure the total mass, m₂.

将液体倒入烧杯，测量总质量，记为 m₂。
The mass of the liquid is m₂ – m₁. Density = (m₂ – m₁) / V.

液体的质量为 m₂ – m₁。密度 = (m₂ – m₁) / V。

One important practical note: for floating objects, such as wood, a thin needle or sinker must be used to push the object completely under the water. Alternatively, the volume can be measured by using a sinker to submerge the object.

实验中需要注意：对于漂浮物（如木块），需要用细针或重物将物体完全压入水中；也可以使用重物使物体完全浸没后再测量体积。

5. Pressure: Definition and Formula | 压强：定义与公式

Pressure is defined as the force acting normally (perpendicular) on a surface per unit area. In Edexcel IGCSE Physics, pressure is usually applied to solid surfaces.

压强定义为单位面积上所受的垂直作用力。在 Edexcel IGCSE 物理中，压强通常适用于固体表面。

The formula for pressure is:

P = F / A

where P is pressure, F is the normal force and A is the area of the surface. The unit of pressure is the pascal (Pa), and 1 Pa = 1 N/m².

其中 P 是压强，F 是垂直于表面的力，A 是表面积。压强的单位是帕斯卡（Pa），且 1 Pa = 1 N/m²。

Pressure can be increased either by increasing the force or by decreasing the area over which the force acts. For example, a sharp knife has a very small contact area, producing a high pressure that easily cuts objects. Snowshoes have a large area to reduce pressure on soft snow and prevent sinking.

增大压强的方法可以是增大压力，或减小受力面积。例如，锋利的刀具有很小的接触面积，产生很高的压强因而容易切割物体；雪鞋则通过增大面积来减小对松软雪地的压强，防止陷入雪中。

6. Pressure in Liquids | 液体压强

Liquids also exert pressure on any surface in contact with them. This pressure is due to the weight of the liquid above and around the point of measurement. The formula for liquid pressure is:

液体也会对与其接触的表面施加压强。这种压强来源于测量点上方和周围液体所受的重力。液体压强的公式为：

P = hρg

where h is the vertical depth below the surface, ρ is the density of the liquid, and g is the gravitational field strength (approximately 10 N/kg in IGCSE calculations, or 9.81 N/kg for more precise work).

其中 h 是距液面的垂直深度，ρ 是液体密度，g 是重力场强度（在 IGCSE 计算中通常取 10 N/kg，更精确时取 9.81 N/kg）。

Key properties of liquid pressure:

液体压强的关键性质：

Liquid pressure increases with depth. The deeper you go, the greater the pressure.

液体压强随深度增加而增大。深度越深，压强越大。
Liquid pressure increases with the density of the liquid. A denser liquid exerts greater pressure at the same depth.

液体压强随液体密度增大而增大。在同一深度，密度更大的液体产生的压强更大。
Liquid pressure acts in all directions, not just downwards. A small hole in the side of a container will release liquid sideways due to lateral pressure.

液体压强向各个方向作用，而不只是向下。容器侧面的小孔会因侧向压强而喷出液体。
At a given depth, the pressure is the same in all directions, regardless of the shape of the container.

在相同深度下，液体压强在各个方向都相等，且与容器的形状无关。

This explains why dam walls are made thicker at the bottom. The pressure against the dam is much greater at greater depths, so the lower sections need more concrete to withstand the force.

这就是为什么水坝的底部要修建得更厚。水对大坝底部的压强远大于浅处，因此设计时需要加强下部结构以承受更大的力。

7. Atmospheric Pressure and the Barometer | 大气压强与气压计

The Earth’s atmosphere is a thick layer of air around the planet. The air has mass, so it exerts pressure on the Earth’s surface. Atmospheric pressure at sea level is approximately 101 kPa, and it decreases with altitude because there is less air above at high altitudes.

地球周围有一层浓厚的大气。空气具有质量，因此会对地球表面产生压强。海平面附近的大气压强约为 101 kPa，并且随着海拔升高而下降，因为高处上方的空气更少。

Atmospheric pressure can be measured using a mercury barometer. A simple mercury barometer consists of a long glass tube filled completely with mercury, then inverted with its open end below the surface of mercury in a bowl. The mercury column falls until the pressure due to the weight of the mercury column balances the atmospheric pressure pressing on the surface of the mercury in the bowl.

大气压强可以用水银气压计测量。简单的水银气压计是一根装满水银的长玻璃管，倒置后开口端浸入水银槽中。水银柱会下降到某一高度，使得水银柱的重力压强与大气压在水银槽表面产生的压强相平衡。

The height of the mercury column is about 760 mm at standard atmospheric pressure. Using P = hρg with h = 0.760 m, ρ = 13600 kg/m³ and g = 9.81 N/kg gives:

标准大气压下，水银柱高度约为 760 mm。利用 P = hρg，取 h = 0.760 m，ρ = 13600 kg/m³，g = 9.81 N/kg，可得：

P = 0.760 × 13600 × 9.81 ≈ 101,000 Pa

An aneroid barometer is another type that uses a small flexible metal box with some air removed. As atmospheric pressure changes, the box expands or contracts, moving a pointer on a scale.

空盒气压计是另一种类型，它使用一个部分抽空气体的柔性金属盒。当大气压强变化时，金属盒会膨胀或收缩，带动指针在刻度盘上移动。

8. The Manometer | 液体压强计（U 形管）

A manometer is a U-shaped tube containing a liquid, usually water or mercury. It is used to measure the pressure of a gas in a container.

液体压强计是一个内装液体（通常是水或水银）的 U 形管，用于测量容器内气体的压强。

One end of the manometer is connected to the gas supply, while the other end is open to the atmosphere. If the gas pressure equals atmospheric pressure, the liquid levels in both arms are the same. If the gas pressure is greater than atmospheric pressure, the liquid is pushed down on the gas side and rises higher on the open side. The difference in liquid height h is a measure of the pressure difference between the gas and the atmosphere:

U 形管的一端连接待测气体，另一端开口通向大气。如果气体压强等于大气压，两臂液面高度相同。如果气体压强大于大气压，液体会在气体侧被压低，而在开口侧上升。液面高度差 h 反映了气体与大气之间的压强差：

P_gas – P_atmosphere = hρg

Therefore the total gas pressure is P_gas = P_atmosphere + hρg.

因此气体总压强为 P_gas = P_atmosphere + hρg。

In an exam question, you may be given the density of the liquid, the height difference h, and atmospheric pressure, and be asked to calculate the gas pressure. Always remember to include atmospheric pressure.

在考试题目中，可能会给出液体密度、高度差 h 和大气压强，要求计算气体压强。请务必记住要把大气压强加进去。

9. Hydraulic Systems and Their Applications | 液压系统及其应用

Liquids are incompressible, and pressure applied to an enclosed liquid is transmitted equally to every part of the liquid. This principle is used in hydraulic machines.

液体不可压缩，并且对封闭液体施加的压强会被等值地传递到液体的各个部分。这一原理被用于液压机械中。

In a simple hydraulic system, there are two pistons with different cross-sectional areas A₁ and A₂. If a force F₁ is applied to the smaller piston, the pressure in the liquid is F₁ / A₁. This same pressure is transmitted to the larger piston, producing a force F₂ given by:

在简单的液压系统中，有两个横截面积不同的活塞 A₁ 和 A₂。如果对小活塞施加力 F₁，液体中的压强就是 F₁ / A₁。这一压强被等值传递到大活塞上，产生力 F₂：

F₁ / A₁ = F₂ / A₂

Rearranging to find the force on the larger piston:

整理后可以求出大活塞上的力：

F₂ = F₁ × A₂ / A₁

Worked example: A₁ = 0.01 m², A₂ = 0.5 m², F₁ = 100 N. Then F₂ = 100 × 0.5 / 0.01 = 5000 N. A small input force is multiplied into a much larger output force.

计算示例：A₁ = 0.01 m²，A₂ = 0.5 m²，F₁ = 100 N。则 F₂ = 100 × 0.5 / 0.01 = 5000 N。这样小的输入力就被放大为很大的输出力。

Hydraulic systems are used in car brakes, hydraulic jacks, and construction machines such as excavators. They allow heavy loads to be lifted with relatively little effort.

液压系统用于汽车制动器、液压千斤顶以及挖掘机等工程机械中。它们可以使重物用较小的力就能被抬升。

10. Exam Tips and Worked Example | 考试技巧与例题

Students often lose marks on density and pressure questions because of unit errors or because they forget to include atmospheric pressure in manometer calculations. Here are a few essential tips.

学生在密度和压强题目中失分，通常是因为单位换算错误，或在 U 形管计算中忘记加上大气压强。以下是几个关键技巧。

Always convert volumes to m³ when using the SI density formula with mass in kg. Remember that 1 cm³ = 1 × 10⁻⁶ m³ and 1 litre = 1000 cm³ = 1 × 10⁻³ m³.

在使用质量（kg）和密度（kg/m³）的 SI 公式时，务必把体积换算为 m³。记住 1 cm³ = 1 × 10⁻⁶ m³，1 升 = 1000 cm³ = 1 × 10⁻³ m³。
State the correct formula before substituting numbers. Show your working clearly because method marks are often awarded even for numerical errors.

代入数值前先写出正确公式。书写要清晰，因为即使计算有误，方法分也常常会被保留。
For liquid pressure, note that P = hρg only gives the pressure due to the liquid. If the question asks for the total pressure at a depth, add the atmospheric pressure above the liquid surface.

注意 P = hρg 只求液体本身产生的压强。如果题目要求某一深度的总压强，还需要加上液面上方的大气压强。
Distinguish between ‘mass’ and ‘weight’. Mass is in kg and weight is a force in N, calculated as W = mg.

区分 ‘质量’ 和 ‘重力’。质量单位是 kg，重力是一种力，单位是 N，计算公式为 W = mg。

Worked example: A cube of ice has side length 0.050 m and mass 0.115 kg. Calculate its density in kg/m³.

例题：一个冰块边长为 0.050 m，质量为 0.115 kg。求它的密度（kg/m³）。

V = side³ = 0.050 × 0.050 × 0.050 = 1.25 × 10⁻⁴ m³. Then ρ = m/V = 0.115 / (1.25 × 10⁻⁴) = 920 kg/m³.

V = 边长³ = 0.050 × 0.050 × 0.050 = 1.25 × 10⁻⁴ m³。则 ρ = m/V = 0.115 / (1.25 × 10⁻⁴) = 920 kg/m³。

Another worked example: A swimming pool is 3.0 m deep. Take ρ_water = 1000 kg/m³ and g = 10 N/kg. Calculate the pressure due to water at the bottom.

另一个例题：游泳池水深 3.0 m。取 ρ_water = 1000 kg/m³，g = 10 N/kg。求池底水产生的压强。

P = hρg = 3.0 × 1000 × 10 = 30,000 Pa = 30 kPa

If atmospheric pressure is 101 kPa, the total pressure at the bottom is 101 + 30 = 131 kPa.

若大气压强为 101 kPa，则池底总压强为 101 + 30 = 131 kPa。

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Substance \| 物质	Density / kg m⁻³ \| 密度 / kg m⁻³
Water \| 水	1000
Ice \| 冰	920
Aluminium \| 铝	2700
Iron \| 铁	7870
Air \| 空气	1.2

Density & Pressure | 密度与压强