Archimedes' Principle and Its Applications | 阿基米德原理及其应用

📚 Archimedes’ Principle and Its Applications | 阿基米德原理及其应用

Why does a heavy steel ship float while a small iron nail sinks? Why do hot air balloons rise into the sky? The answer lies in one of the oldest and most elegant laws of physics: Archimedes’ Principle. This principle explains the upward force we call buoyancy and governs countless phenomena in everyday life, from swimming to shipbuilding.

为什么一艘沉重的钢铁轮船能漂浮在水面上，而一枚小小的铁钉却沉入水底？为什么热气球能升上天空？答案在于物理学中最古老、最优雅的定律之一——阿基米德原理。这一定律解释了被称为浮力的向上作用力，并支配着从游泳到造船等日常生活中的无数现象。

1. What Is Buoyancy? | 什么是浮力？

When an object is placed in a fluid (a liquid or a gas), the fluid exerts an upward force on the object. This upward force is called the buoyant force. It exists because pressure in a fluid increases with depth: the bottom of the object experiences higher pressure than the top, creating a net upward push.

当一个物体被放入流体（液体或气体）中时，流体会对物体施加一个向上的力。这个向上的力称为浮力。浮力之所以存在，是因为流体中的压力随深度增加而增大：物体底部受到的压力大于顶部，从而产生一个净向上的推力。

Interestingly, the buoyant force does not depend on the shape or material of the object alone. It depends only on how much fluid the object displaces. A fully submerged object displaces its own volume of fluid; a floating object displaces only the volume that is below the surface.

有趣的是，浮力并不单独取决于物体的形状或材料。它只取决于物体排开了多少流体。完全浸没的物体排开自身体积的流体；漂浮的物体只排开位于液面以下的那部分体积。

2. Statement of Archimedes’ Principle | 阿基米德原理的表述

Archimedes’ Principle states: When an object is wholly or partially immersed in a fluid, the fluid exerts an upward buoyant force on the object equal to the weight of the fluid displaced by the object.

阿基米德原理指出：当物体全部或部分浸没在流体中时，流体对物体施加的向上浮力等于物体所排开的流体所受的重力。

This principle was discovered by the Greek mathematician Archimedes over 2,000 years ago. Legend has it that he realised the concept while taking a bath and famously ran through the streets shouting “Eureka!” (I have found it!).

这一原理由希腊数学家阿基米德在两千多年前发现。传说他在洗澡时领悟了这一概念，并赤身跑上大街高喊“尤里卡！”（我找到了！）。

The principle applies to all fluids, not just water. It works for gases too, which is why helium balloons float in air. The buoyant force is always equal to the weight of the displaced fluid, regardless of the object’s composition.

该原理适用于所有流体，不仅限于水，也适用于气体。这就是氦气球能在空气中漂浮的原因。浮力始终等于被排开流体的重量，与物体的成分无关。

3. The Mathematical Formula | 数学公式

The buoyant force can be expressed mathematically as:

浮力可以用数学公式表达为：

F_b = m_f × g = ρ_f × V_f × g

where F_b is the buoyant force in newtons (N), m_f is the mass of the displaced fluid (kg), ρ_f is the density of the fluid (kg/m³), V_f is the volume of fluid displaced (m³), and g is the gravitational field strength (approximately 9.8 N/kg on Earth).

其中 F_b 是浮力，单位为牛顿（N）；m_f 是被排开流体的质量（kg）；ρ_f 是流体的密度（kg/m³）；V_f 是被排开流体的体积（m³）；g 是重力场强度（地球上约为 9.8 N/kg）。

Since mass equals density multiplied by volume, the formula shows that a denser fluid produces a larger buoyant force for the same displaced volume. This is why it is easier to float in seawater (density ≈ 1,025 kg/m³) than in fresh water (density ≈ 1,000 kg/m³).

由于质量等于密度乘以体积，公式表明在排开体积相同的情况下，密度更大的流体产生更大的浮力。这就是为什么在海水（密度约 1,025 kg/m³）中比在淡水（密度约 1,000 kg/m³）中更容易漂浮。

4. Floating, Sinking, and Hovering | 漂浮、下沉与悬浮

Comparing the buoyant force F_b with the object’s weight W determines the object’s behaviour in a fluid:

比较浮力 F_b 与物体重力 W 的大小，可以判断物体在流体中的行为：

F_b > W: The net force is upward, so the object accelerates upward and eventually floats on the surface.

F_b > W：净力向上，物体向上加速，最终漂浮在液面上。
F_b = W: The object is in equilibrium. It can remain suspended at any depth (hovering).

F_b = W：物体处于平衡状态，可以在任意深度保持悬浮。
F_b < W: The net force is downward, so the object sinks to the bottom.

F_b < W：净力向下，物体下沉到底部。

For a fully submerged object, the buoyant force is fixed at F_b = ρ_f × V_object × g. Therefore, the density of the object compared to the density of the fluid is the real deciding factor:

对于完全浸没的物体，浮力恒定为 F_b = ρ_f × V_object × g。因此，物体密度与流体密度的比较才是真正的决定因素：

If ρ_object < ρ_f, the object floats.

若 ρ_物体 < ρ_流体，物体上浮。
If ρ_object = ρ_f, the object hovers.

若 ρ_物体 = ρ_流体，物体悬浮。
If ρ_object > ρ_f, the object sinks.

若 ρ_物体 > ρ_流体，物体下沉。

5. Fraction Submerged and Density Relationship | 浸没比例与密度的关系

For a floating object, the buoyant force balances the weight exactly. This gives a powerful relationship:

对于漂浮的物体，浮力恰好平衡重力。这给出了一个非常有用的关系式：

ρ_f × V_submerged × g = ρ_object × V_object × g

After cancelling g from both sides, we obtain:

两边同时消去 g 后，得到：

V_submerged / V_object = ρ_object / ρ_f

This formula tells us the fraction of the object’s volume that sits below the surface. For example, an iceberg with density 917 kg/m³ floating in seawater (1,025 kg/m³) has a submerged fraction of 917/1,025 ≈ 0.895, meaning about 90% of the iceberg is under water. This is why “only the tip of the iceberg” is visible.

这个公式告诉我们物体浸没在水面以下的体积比例。例如，密度为 917 kg/m³ 的冰山漂浮在海水（1,025 kg/m³）中，浸没比例为 917/1,025 ≈ 0.895，意味着约 90% 的冰山位于水下。这就是为什么“只见冰山一角”。

6. Apparent Weight and Loss of Weight | 视重与失重

When an object is immersed in a fluid, its measured weight decreases. This is called the apparent weight:

当物体浸入流体中时，其测得的重量会减小。这被称为视重：

W_apparent = W − F_b

This principle is used to measure density. An object weighing 30 N in air may weigh only 20 N in water. The loss of 10 N equals the buoyant force, which equals the weight of the displaced water (10 N). From this, the volume and density of the object can be calculated.

这一原理可用于测量密度。一个在空气中重 30 N 的物体，在水中可能只重 20 N。减少的 10 N 等于浮力，也等于被排开水的重量（10 N）。由此可以计算出物体的体积和密度。

This is the basis of the classic “weighing method” for finding the density of irregular objects. It connects the concept of density to measurable weight loss in a very practical way.

这是经典“称量法”测定不规则物体密度的基础。它以非常实用的方式将密度概念与可测量的失重联系起来。

7. Application: Ships and Steel Boats | 应用：轮船与钢铁船只

Steel has a density of about 7,800 kg/m³, far greater than water’s 1,000 kg/m³, so a solid block of steel sinks. However, a ship is built with a hollow hull. The overall density of the ship, including its cargo and trapped air, is less than the density of water.

钢铁的密度约为 7,800 kg/m³，远大于水的 1,000 kg/m³，因此实心钢块会下沉。然而，船体被制造成中空的。包括货物和内部空气在内的整船平均密度小于水的密度。

A ship floats when the weight of the water it displaces equals the total weight of the ship. The larger the submerged volume, the greater the buoyant force. Ship builders use this principle to design hulls that displace enough water to support heavy loads.

当船排开水的重量等于整艘船的重量时，船就能漂浮。浸没体积越大，浮力就越大。造船工程师利用这一原理设计船体，使其排开足够多的水来支撑沉重载荷。

The maximum load a ship can carry is determined by the difference between the maximum buoyant force and the ship’s own weight. This is the ship’s load capacity, often indicated by the Plimsoll line painted on the hull.

船只的最大载重量由最大浮力与船自身重量之差决定，这就是船的载重能力，通常由绘制在船体上的载重线（Plimsoll 线）标示。

8. Application: Submarines | 应用：潜艇

Submarines control their density by taking in or pushing out water from ballast tanks. When the tanks are filled with seawater, the submarine’s average density increases and it sinks. When the tanks are filled with compressed air to expel water, the average density decreases and the submarine rises.

潜艇通过压载水舱进水和排水来控制自身密度。当水舱充满海水时，潜艇的平均密度增大而下沉。当用压缩空气将水排出时，平均密度减小，潜艇上浮。

By adjusting the amount of water in the tanks, the crew can achieve neutral buoyancy. In this state, F_b equals W exactly, and the submarine can hover silently at a constant depth using only its horizontal fins to manoeuvre.

通过调节水舱中的水量，船员可以实现中性浮力。在这种状态下，浮力恰好等于重力，潜艇可以在恒定深度安静地悬停，仅依靠水平鳍进行机动。

9. Application: Hydrometers and Measuring Density | 应用：比重计与密度测量

A hydrometer is an instrument used to measure the density of liquids. It consists of a sealed glass tube with a weighted bulb at one end. When placed in a liquid, it floats upright. The depth to which it sinks depends on the liquid’s density.

比重计是一种用于测量液体密度的仪器。它由一根密封的玻璃管组成，一端带有重球。放入液体后，它会竖直漂浮。下沉的深度取决于液体的密度。

In a denser liquid, the hydrometer requires less volume submerged to displace the same weight of liquid, so it floats higher. In a less dense liquid, it sinks deeper. A calibrated scale on the tube allows direct reading of the liquid’s density.

在密度较大的液体中，比重计只需排开较少体积即可产生等于自身重量的浮力，因此浮得更高。在密度较小的液体中则下沉更深。管上的校准刻度可以直接读出液体的密度。

Hydrometers are widely used in industry, from testing the charge level of car batteries (by measuring the density of sulfuric acid) to checking the sugar content in wine and beer production.

比重计在工业中应用广泛，从检测汽车电池的电量（通过测量硫酸密度）到检查葡萄酒和啤酒生产中的糖分含量，都离不开它。

10. Application: Hot Air Balloons and Airships | 应用：热气球与飞艇

Archimedes’ Principle applies to gases as well. A balloon filled with helium (density ≈ 0.179 kg/m³) experiences a buoyant force equal to the weight of the air it displaces. Since ambient air has a density of about 1.225 kg/m³ at sea level, the buoyant force is larger than the weight of the helium, producing a net upward lift.

阿基米德原理同样适用于气体。充满氦气（密度约 0.179 kg/m³）的气球会受到等于其所排开空气重量的浮力。由于海平面附近空气的密度约为 1.225 kg/m³，浮力大于氦气本身的重力，从而产生净向上的升力。

Hot air balloons work the same way. Burning propane heats the air inside the envelope, lowering its density. When the average density of the balloon (including the hot air, basket, and passengers) becomes less than the surrounding cold air, the balloon rises. Cooling the air, or venting some of it, makes the balloon descend.

热气球的工作原理相同。燃烧丙烷加热球囊内的空气，使其密度降低。当气球（包括热空气、吊篮和乘客）的平均密度小于周围冷空气时，气球上升。冷却空气或排出部分空气则使气球下降。

11. Worked Example | 例题解析

Problem: A rectangular block of wood has a density of 600 kg/m³ and a volume of 0.005 m³. It is placed in fresh water (density 1,000 kg/m³). Calculate (a) the weight of the block, (b) the volume of water displaced, and (c) the fraction of the block submerged.

题目：一块长方体木头密度为 600 kg/m³，体积为 0.005 m³。将其放入淡水（密度 1,000 kg/m³）中。计算（a）木块的重力；（b）排开水的体积；（c）木块浸没的比例。

Solution:

解答：

(a) Mass of the block = ρ × V = 600 × 0.005 = 3 kg. Weight = m × g = 3 × 9.8 = 29.4 N.

（a）木块质量 = ρ × V = 600 × 0.005 = 3 kg。重力 = m × g = 3 × 9.8 = 29.4 N。

(b) For floating, F_b = W = 29.4 N. Since F_b = ρ_water × V_displaced × g, we have 29.4 = 1,000 × V_displaced × 9.8. Therefore V_displaced = 29.4 / (1,000 × 9.8) = 0.003 m³.

（b）漂浮时，F_b = W = 29.4 N。由 F_b = ρ_水 × V_排 × g，可得 29.4 = 1,000 × V_排 × 9.8。因此 V_排 = 29.4 / (1,000 × 9.8) = 0.003 m³。

(c) Fraction submerged = V_submerged / V_object = 0.003 / 0.005 = 0.6, or 60%. This matches the density ratio: 600 / 1,000 = 0.6.

（c）浸没比例 = V_浸没 / V_物体 = 0.003 / 0.005 = 0.6，即 60%。这与密度比一致：600 / 1,000 = 0.6。

12. Common Misconceptions | 常见误区

Misconception 1: “An object sinks because it is heavy.” In reality, sinking depends on density, not just weight. A heavy log floats while a light nail sinks because the log’s density is lower than water’s.

误区一：“物体下沉是因为它重。”实际上，下沉取决于密度，而不仅仅是重量。沉重的圆木漂浮而轻小的铁钉下沉，因为圆木的密度小于水。

Misconception 2: “The buoyant force increases with depth.” In fact, for a fully submerged object, the buoyant force is constant regardless of depth, because the density of the fluid and the volume displaced do not change. Pressure increases with depth, but the pressure difference between top and bottom remains the same.

误区二：“浮力随深度增加而增大。”事实上，对于完全浸没的物体，浮力恒定不变，与深度无关，因为流体密度和排开体积都不改变。压强随深度增加，但上下表面间的压强差保持不变。

Misconception 3: “Objects float better in a vacuum.” No, vacuum has no fluid and therefore no buoyancy. In fact, a helium balloon only rises because the surrounding air provides an upward buoyant force.

误区三：“物体在真空中漂浮得更好。”不对，真空没有流体，因此没有浮力。实际上，氦气球之所以上升，正是因为它周围的空气提供了向上的浮力。

Misconception 4: “A ship floating on water experiences no buoyant force.” This is false — the ship is in equilibrium, meaning the buoyant force exactly equals the ship’s weight. Without buoyancy, the ship would sink.

误区四：“漂浮在水面上的船不受浮力作用。”这是错误的——船处于平衡状态，意味着浮力恰好等于船的重力。没有浮力，船就会沉底。

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Archimedes’ Principle and Its Applications | 阿基米德原理及其应用