The Gravitational Force Formula G=mg and Its Applications | 重力公式G=mg及其应用

📚 The Gravitational Force Formula G=mg and Its Applications | 重力公式G=mg及其应用

The equation G=mg is one of the first and most fundamental relationships introduced in A-level Physics. It connects the gravitational force acting on an object to its mass and the local gravitational field strength, and it serves as a springboard for understanding weight, free fall, projectile motion, and even satellite dynamics.

公式 G=mg 是 A-level 物理中最基础也是最先引入的关系之一。它将物体所受的重力与其质量及当地重力场强度联系起来，是理解重量、自由落体、抛体运动乃至卫星运动的重要起点。

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1. Defining Weight and Gravitational Field Strength | 定义重量与重力场强度

In physics, weight is not the same as mass. Mass is a measure of the amount of matter in an object, measured in kilograms (kg), and is invariant regardless of location. Weight, however, is the gravitational force exerted on that mass by a planet or other celestial body, measured in newtons (N).

物理学中，重量与质量并不相同。质量是物体所含物质的量，单位为千克（kg），与所处位置无关。而重量是行星或其他天体对该质量施加的引力，单位为牛顿（N）。

The gravitational field strength g represents the force acting per unit mass at a given point in a gravitational field. On the surface of the Earth, g is approximately 9.81 N kg⁻¹ (or equivalently 9.81 m s⁻²), but this value varies slightly with altitude, latitude, and local geology.

重力场强度 g 表示重力场中某一点处每单位质量所受的力。在地球表面，g 约为 9.81 N kg⁻¹（也等价于 9.81 m s⁻²），但该值会随海拔高度、纬度和局部地质结构而略有变化。

Weight = mass × gravitational field strength

重量 = 质量 × 重力场强度

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2. Deriving G=mg and Its Vector Nature | 推导 G=mg 及其矢量性质

The formula G=mg is a direct application of Newton’s Second Law, F=ma. When the only acceleration acting on a freely falling object is due to gravity, the acceleration a is replaced by g, and the force F becomes the weight G. Thus, Newton’s Second Law reduces to G=mg.

公式 G=mg 是牛顿第二定律 F=ma 的直接应用。当自由落体物体仅受重力作用时，加速度 a 被替换为 g，力 F 即为重量 G，牛顿第二定律便简化为 G=mg。

It is important to recognise that both G and g are vector quantities, directed towards the centre of the Earth. When solving problems, it is common practice to assign a negative sign to the upward direction if downward is taken as positive, or vice versa, to keep calculations consistent.

需注意 G 和 g 都是矢量，方向指向地心。解题时通常取一个方向为正，如取向下为正，则向上为负，以保持计算的一致性。

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3. Variation of g with Location | g 随位置的变化

The gravitational field strength g is not constant across the universe, or even across the surface of the Earth. It depends on the mass of the celestial body and the distance from its centre according to the relation:

重力场强度 g 并非在宇宙中恒定，甚至在地球表面也不是恒定值。它由天体的质量以及到天体中心的距离决定，其关系式为：

g = G M / r²

where G is the universal gravitational constant (6.67 × 10⁻¹¹ N m² kg⁻²), M is the mass of the planet, and r is the distance from the planet’s centre. On the Earth’s surface, r equals the Earth’s radius R; at a height h above the surface, r becomes R + h, leading to a smaller g.

其中 G 为万有引力常量（6.67 × 10⁻¹¹ N m² kg⁻²），M 为星球质量，r 为到星球中心的距离。在地球表面，r 等于地球半径 R；在距地面高度 h 处，r 变为 R + h，导致 g 减小。

• At sea level: g ≈ 9.81 m s⁻²
• At the top of Mount Everest (h ≈ 8.8 km): g ≈ 9.77 m s⁻²
• Inside a deep mine: g decreases as less mass lies below the object

• 海平面处：g ≈ 9.81 m s⁻²
• 珠穆朗玛峰顶（h ≈ 8.8 km）：g ≈ 9.77 m s⁻²
• 深矿井内：g 减小，因为物体下方质量减少

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4. Mass vs. Weight: A Common Pitfall | 质量与重量：常见易错点

A frequent source of confusion in exams is distinguishing between mass and weight. Mass is a scalar quantity measured in kilograms and remains unchanged in any location. Weight is a force, measured in newtons, and depends on the local value of g.

考试中一个常见的混淆点是区分质量与重量。质量是标量，单位为千克，在任何地方都不变；重量是力，单位为牛顿，取决于当地的 g 值。

For example, an astronaut with a mass of 70 kg weighs approximately 687 N on Earth (70 × 9.81). On the Moon, where g ≈ 1.62 m s⁻², the same astronaut weighs only about 113 N, yet their mass remains 70 kg. This distinction is crucial when applying G=mg in different contexts.

例如，一位质量为 70 kg 的宇航员在地球上重量约为 687 N（70 × 9.81）。在月球上，g ≈ 1.62 m s⁻²，同样这位宇航员仅重约 113 N，但质量仍为 70 kg。在不同情境下应用 G=mg 时，这一区分至关重要。

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5. Applications in Free Fall and Vertical Motion | 在自由落体和竖直运动中的应用

When an object falls freely under gravity, the only force acting on it is its weight G=mg. By applying Newton’s Second Law:

当物体在重力作用下自由下落时，唯一作用力是重力 G=mg。根据牛顿第二定律：

ma = mg, therefore a = g

This means that all objects in free fall experience the same acceleration, regardless of their mass. This principle leads directly to the equations of uniformly accelerated motion commonly used in A-level calculations:

这意味着所有自由落体物体无论质量大小，都经历相同的加速度。这一原理直接导出 A-level 计算中常用的匀加速运动方程：

v = u + gt

s = ut + ½gt²

v² = u² + 2gs

Where u is the initial velocity, v the final velocity, s the displacement, and t the time. In these equations, g is used as the acceleration, and proper sign conventions must be maintained.

其中 u 为初速度，v 为末速度，s 为位移，t 为时间。在这些方程中，g 作为加速度使用，须保持正确的正负号约定。

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6. Projectile Motion under Gravity | 重力作用下的抛体运动

In projectile motion, an object is given an initial velocity at an angle to the horizontal. The horizontal component of velocity remains constant (ignoring air resistance), while the vertical component is affected by g. The weight G=mg provides the vertical acceleration g throughout the flight.

在抛体运动中，物体以与水平方向成某角度的初速度抛出。水平速度分量保持不变（忽略空气阻力），而竖直分量受 g 影响。重力 G=mg 在整个飞行过程中提供竖直加速度 g。

Horizontal motion: aₓ = 0, vₓ = u cos θ, sₓ = (u cos θ)t

竖直运动：a_y = −g, v_y = u sin θ − gt, s_y = (u sin θ)t − ½gt²

These equations allow calculation of the time of flight, maximum height, and range of a projectile. Examiners often test the understanding that the horizontal and vertical motions are independent, connected only by time t.

这些方程可用于计算飞行时间、最大高度和射程。考官常考查对水平与竖直运动相互独立、仅由时间 t 联系的理解。

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7. Relationship between G, m, and g in Equilibrium | 平衡状态下 G、m 与 g 的关系

When an object is at rest on a horizontal surface, its weight G=mg acts vertically downward, and the surface exerts an equal and opposite normal reaction force R. In equilibrium, the net vertical force is zero:

当物体在水平面上静止时，其重力 G=mg 竖直向下作用，表面施加大小相等、方向相反的法向反作用力 R。在平衡状态下，竖直方向合力为零：

R = mg

This simple relationship is widely used in problems involving weighing scales, spring balances, and calculating normal reaction forces on inclines. On an inclined plane at angle θ to the horizontal, the component of weight perpendicular to the plane is mg cos θ, and the component parallel to the plane is mg sin θ.

这一简单关系广泛应用于涉及称重秤、弹簧秤以及斜面上法向反作用力的计算问题中。在与水平面成 θ 角的斜面上，重力垂直于斜面的分量为 mg cos θ，平行于斜面的分量为 mg sin θ。

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8. Experimental Determination of g | 重力加速度 g 的实验测定

In the laboratory, the value of g can be determined using a simple pendulum or a free-fall apparatus. For a simple pendulum of length L, the period T is related to g by:

在实验室中，可采用单摆或自由落体装置来测定 g 值。对于长度为 L 的单摆，其周期 T 与 g 的关系为：

T = 2π √(L / g)

Rearranging gives g = 4π²L / T², and by measuring T for various L and plotting a graph of T² against L, the gradient yields g. This method is a classic A-level practical, testing skills in measurement, graphing, and error analysis.

整理得 g = 4π²L / T²，通过测量不同 L 对应的 T，并绘制 T² 对 L 的图线，可利用斜率求得 g。该方法是 A-level 经典实验，考查测量、作图及误差分析技能。

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9. Apparent Weight and Elevator Problems | 视重与电梯问题

When an object is placed in an accelerating elevator, the normal reaction force R (the apparent weight) differs from mg. This can be analysed using Newton’s Second Law. For an elevator accelerating upwards with acceleration a:

当物体位于加速上升的电梯中时，法向反作用力 R（即视重）与 mg 不同。可通过牛顿第二定律分析。对于以加速度 a 向上加速的电梯：

R − mg = ma, so R = m(g + a)

For an elevator accelerating downwards:

对于向下加速的电梯：

mg − R = ma, so R = m(g − a)

In free fall (a = g), R becomes zero, leading to apparent weightlessness. This concept links G=mg to the phenomenon of weightlessness experienced by astronauts and is a common exam topic.

在自由落体（a = g）情况下，R 变为零，出现视重为零，即“失重”状态。这一概念将 G=mg 与宇航员经历的失重现象联系起来，是常见考点。

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10. Gravitational Potential Energy and G=mg | 重力势能与 G=mg

Near the Earth’s surface, where g is approximately constant, the gravitational potential energy of an object of mass m at height h is given by:

在地球表面附近，g 近似恒定，质量为 m 的物体在高度 h 处的重力势能为：

Eₚ = mgh

This formula is derived from the work done against the weight G=mg to raise the object through a height h. The work done equals force multiplied by distance, so W = mg × h. This energy relationship is widely applied in problems involving falling objects, roller coasters, and hydroelectric power generation.

该公式源于克服重力 G=mg 将物体提升高度 h 所做的功。功等于力乘以距离，即 W = mg × h。这一能量关系广泛应用于涉及落体、过山车以及水力发电的问题中。

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11. Satellites and Weightlessness | 卫星与失重状态

For a satellite in a circular orbit at distance r from the Earth’s centre, the gravitational force G=mg provides the centripetal force required for circular motion:

对于在距地心 r 处做圆周运动的卫星，重力 G=mg 提供圆周运动所需的向心力：

mg = mv² / r

Cancelling m gives the orbital speed v = √(gr). Since g = GM / r², this becomes v = √(GM / r). Astronauts in orbit appear weightless because they are in a continuous state of free fall around the Earth — their acceleration is exactly g, and there is no normal reaction force from a supporting surface.

约去 m 得轨道速度 v = √(gr)。由于 g = GM / r²，此式变为 v = √(GM / r)。轨道中的宇航员看起来“失重”，是因为他们处于绕地球的持续自由落体状态——其加速度恰为 g，且没有来自支撑面的法向反作用力。

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12. Summary and Key Exam Tips | 总结与关键备考提示

A clear grasp of G=mg is essential for success in A-level Physics. Start by memorising the relationship between weight, mass, and gravitational field strength, and practise applying it across diverse contexts: free fall, projectiles, inclined planes, elevators, and orbital motion.

熟练掌握 G=mg 对 A-level 物理的成功至关重要。首先要牢记重量、质量与重力场强度之间的关系，并在多种情境中练习应用：自由落体、抛体、斜面、电梯和轨道运动。

Quantity	Symbol	Unit	Scalar/Vector
Mass	m	kg	Scalar
Weight	G	N	Vector
Gravitational field strength	g	N kg⁻¹ / m s⁻²	Vector

物理量	符号	单位	标量/矢量
质量	m	kg	标量
重量	G	N	矢量
重力场强度	g	N kg⁻¹ / m s⁻²	矢量

In any exam problem, first identify all forces acting on the object, apply Newton’s Second Law, and then use G=mg to calculate the weight. Always quote final answers with correct units and direction. Pay attention to the value of g provided in the question — use 9.81 m s⁻² unless otherwise stated.

在答任何题目时，首先确定物体的全部受力，应用牛顿第二定律，然后使用 G=mg 计算重量。最终答案务必注明正确单位和方向。注意题中给出的 g 值——未注明时一般取 9.81 m s⁻²。

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