📚 A-Level Physics Circular Motion, Centripetal Force and Gravitational Fields | A-Level物理 圆周运动、向心力与引力场考点
Circular motion and gravitational fields are two of the most heavily examined areas of A-Level Physics, and they are also where candidates lose marks most easily. The two topics are deliberately linked in the syllabus: the same centripetal force equation that describes a ball on a string also explains why a satellite stays in orbit, and the same inverse-square law that governs the field of a planet also governs the field of a point charge in your later electrostatics work. This guide walks through definitions, derivations, standard situations and examiner traps step by step.
圆周运动与引力场是A-Level物理中考查频率最高、失分也最集中的两大板块。考纲刻意把两者放在一起：描述绳系小球圆周运动的向心力公式，同样能解释卫星为何能稳定在轨运行；控制行星引力场的平方反比规律，在你后续学习静电场时会再次出现。本文将按定义、推导、典型情境与考场易错点逐层梳理。
1. Describing Circular Motion: Angular Displacement, Period and Frequency | 描述圆周运动：角位移、周期与频率
Circular motion is motion along a circular path at constant speed. The crucial point is that although the speed is constant, the velocity is not: the direction of travel changes continuously, so the velocity vector changes, so the object accelerates. Angular displacement θ is the angle swept out at the centre of the circle, measured in radians. One radian is the angle subtended at the centre when the arc length equals the radius. The period T is the time for one complete revolution, the frequency f is the number of revolutions per second, and the angular speed ω is the rate of change of angular displacement.
圆周运动是物体以恒定速率沿圆形轨道运动。关键点在于：速率虽然不变，速度却在变——运动方向不断改变，速度矢量随之改变，因此物体一定存在加速度。角位移 θ 是物体在圆心处扫过的角度，单位是弧度。1 弧度定义为弧长等于半径时所对应的圆心角。周期 T 是完成一整圈所需时间，频率 f 是每秒转过的圈数，角速度 ω 则是角位移的变化率。
θ = s / r（弧度定义） f = 1 / T ω = Δθ / Δt = 2π / T = 2πf
Remember that ω must be in rad s⁻¹ before you substitute into any equation. A very common error is to leave a rotation rate in revolutions per minute and use it directly, which produces answers wrong by a factor of 2π or 60.
切记代入公式前 ω 必须以 rad s⁻¹ 为单位。最常见的错误就是把"每分钟转数"直接代入，结果会差 2π 倍或 60 倍，整道大题连带失分。
2. Linear Speed and the Relation v = ωr | 线速度与 v = ωr 关系
ime T, the linear speed is v = 2πr / T. Combining this with ω = 2π / T gives the fundamental relation between linear and angular quantities, v = ωr. This equation only holds when ω is measured in rad s⁻¹.
圆周上一点的线速度 v 是单位时间内走过的弧长。转一整圈的路程为 2πr，用时 T，因此 v = 2πr / T。与 ω = 2π / T 联立，即可得到线量与角量的基本关系式 v = ωr。使用该式时 ω 必须以 rad s⁻¹ 为单位。
v = 2πr / T = ωr
For a rigid rotating body such as a wheel or a turntable, every point has the same ω but different v: points further from the axis move faster. For two objects joined by a belt or in mesh, the linear speeds at the contact are equal while the angular speeds differ. The direction of v is always tangential to the circle, perpendicular to the radius at that instant, and it changes continuously.
对车轮、转盘这类刚体，各点 ω 相同但 v 不同：离轴越远，线速度越大。若两轮通过皮带连接或齿轮啮合，则接触点线速度相同而角速度不同。v 的方向始终沿圆周切线，与该点半径垂直，并且不断改变。
3. Centripetal Acceleration: a = v² / r = ω²r | 向心加速度
irected towards the centre of the circle and is called centripetal acceleration. It can be derived by considering the velocity vectors at two instants separated by a small time Δt: the two velocity vectors have the same length v but slightly different directions, and the vector triangle formed is similar to the triangle formed by the two radii and the arc. The change in velocity Δv has magnitude vΔθ, so the acceleration has magnitude vΔθ / Δt = vω.
由于速度方向不断改变，做匀速圆周运动的物体即使速率不变也存在加速度。该加速度指向圆心，称为向心加速度。推导方法是取相隔极短时间 Δt 的两个速度矢量：两者大小均为 v 但方向略有不同，其矢量三角形与"两半径加弧长"构成的三角形相似，速度变化量的大小为 vΔθ，因此加速度大小为 vΔθ / Δt = vω。
a = v² / r = ω²r = vω = (4π²r) / T²
The acceleration is always perpendicular to the velocity, so it changes only the direction of motion, never the speed. If the acceleration had a component along the velocity, the speed would change and the motion would no longer be uniform circular motion. This is why work done by the centripetal force over any complete revolution is zero.
向心加速度始终与速度垂直，因此只改变运动方向，不改变速率。若加速度沿速度方向存在分量，速率就会改变，运动便不再是匀速圆周运动。这也解释了为什么向心力绕一整圈所做的功为零。
4. Centripetal Force and Newton's Second Law | 向心力与牛顿第二定律
By Newton's second law, an acceleration requires a resultant force in the same direction. The resultant force directed towards the centre is called the centripetal force, and it is not a new kind of force: it is simply the name we give to whatever real force (or combination of forces) happens to act towards the centre. Depending on the situation, the centripetal force may be provided by tension, friction, the normal contact force, gravitational attraction, electrostatic force, or a component of any of these.
根据牛顿第二定律，有加速度必有同方向的合力。指向圆心的合力称为向心力，它并不是一种新的力，而是我们给"恰好指向圆心的那个真实力（或若干力的合力）"所起的名字。视情境不同，向心力可以由张力、摩擦力、支持力、万有引力、静电力，或其中某个分量来提供。
F = ma = mv² / r = mω²r = mωv
Two exam conventions matter here. First, never draw a separate arrow labelled "centripetal force" alongside the real forces - that double counting loses marks. Second, "centrifugal force" is not a real force in an inertial frame; it is a useful fiction only in a rotating frame, and A-Level mark schemes generally reject it unless the question explicitly asks about rotating frames.
考场上两条"铁律"必须记住。第一，不要在受力图上额外画一个标着"向心力"的箭头——真实力已经提供了向心效果，重复标注会被扣分。第二，"离心力"在惯性系中不是真实存在的力，它只在转动参考系中作为虚拟力使用；除非题目明确讨论转动参考系，A-Level 评分标准一般不接受这种说法。
5. Standard Situations: Conical Pendulum and Horizontal Circle | 典型情境：锥摆与水平圆周运动
In a conical pendulum, a mass attached to a string of length L moves in a horizontal circle while the string makes a constant angle θ with the vertical. Resolving vertically, the vertical component of tension balances weight; resolving horizontally, the horizontal component supplies the centripetal force. Dividing one equation by the other eliminates the tension and gives a clean expression for the angle.
锥摆模型中，质量为 m 的小球系在长为 L 的绳上，在水平面内做圆周运动，绳与竖直方向保持固定夹角 θ。竖直方向分解：张力的竖直分量与重力平衡；水平方向分解：张力的水平分量提供向心力。两式相除消去张力，即可得到与夹角相关的简洁表达式。
T cos θ = mg T sin θ = mv² / r tan θ = v² / (rg) r = L sin θ
The same technique applies to a car going round a flat bend, where friction provides the centripetal force. Setting the maximum friction equal to the required centripetal force gives the maximum safe speed. Notice that the mass cancels, so a heavy lorry and a small car have the same maximum cornering speed on the same road with the same tyres.
同样的方法适用于汽车在水平弯道上转弯的情形，此时由静摩擦力提供向心力。令最大静摩擦力等于所需向心力，即可求出最大安全速率。注意质量被约去，因此在同一路面、同样轮胎条件下，重卡与小轿车的最大转弯速率相同。
μmg = mv² / r → vmax = √μrg
6. Vertical Circular Motion and the Minimum Speed Condition | 竖直圆周运动与最小速率条件
In vertical circular motion the speed is not constant, so the motion is circular but not uniform. At the top of the loop both the weight and the tension act downwards, towards the centre; at the bottom the tension acts upwards while the weight acts downwards. The critical condition occurs at the top, where the minimum speed corresponds to the tension (or the normal contact force) falling to zero, so that gravity alone supplies the centripetal force.
竖直圆周运动中速率并不恒定，因此它是圆周运动但不是匀速圆周运动。在最高点，重力与张力都指向圆心（向下）；在最低点，张力向上而重力向下。临界条件出现在最高点：当张力（或支持力）恰好减小到零、仅由重力提供向心力时，对应最小速率。
最高点：T + mg = mv² / r → vmin = √gr（T = 0）
最低点：T − mg = mv² / r
If the speed at the top is less than √gr and bottom: ½mv_top² + mg(2r) = ½mv_bottom², which gives a minimum speed at the bottom of √5gr for a complete loop.
若最高点速率小于 √gr，绳会松弛，物体脱离圆轨道而变为抛体运动。用机械能守恒可把最高点与最低点速率联系起来：½mv_顶² + mg(2r) = ½mv_底²，由此得到完成整圈所需的最低点最小速率为 √5gr。
7. Banked Tracks and Cornering at the Design Speed | 倾斜弯道与设计速率
On a banked track the surface is tilted so that the normal contact force itself has a horizontal component pointing towards the centre of curvature. At the design speed, no friction is needed at all and the centripetal force is supplied entirely by the horizontal component of the normal force. This is how motorway curves and velodrome tracks are engineered.
倾斜弯道的路面被垫成一定坡度，使支持力本身具有指向圆心的水平分量。在设计速率下，完全不需要摩擦力，向心力全部由支持力的水平分量提供。高速公路弯道与赛车场自行车赛道就是按这个原理设计的。
N cos θ = mg N sin θ = mv² / r → tan θ = v² / (rg)
If the car travels faster than the design speed it tends to slide up the bank and friction acts down the slope; if it travels more slowly it tends to slide down and friction acts up the slope. In an exam, read carefully whether friction is stated to be negligible or at its limiting value before writing down equations.
若车速高于设计速率，车有沿坡面上滑的趋势，摩擦力沿坡面向下；若车速低于设计速率，车有下滑趋势，摩擦力沿坡面向上。答题前务必看清题干是否说明摩擦力忽略不计或已达到最大值，再决定方程的写法。
8. Newton's Law of Universal Gravitation | 万有引力定律
Newton's law of gravitation states that every point mass attracts every other point mass with a force proportional to the product of their masses and inversely proportional to the square of the distance between their centres. The force acts along the line joining the two centres and forms a Newton's-third-law pair: the two bodies pull each other with equal magnitudes and opposite directions.
牛顿万有引力定律指出：任意两个质点之间都存在引力，其大小与两质点质量的乘积成正比，与两质点中心距离的平方成反比。引力沿两中心的连线方向，并构成一对牛顿第三定律作用力与反作用力：两个物体互相吸引，大小相等、方向相反。
F = GMm / r²，G = 6.67 × 10⁻¹¹ N m² kg⁻²
The law applies exactly to point masses and, as Newton proved, to uniform spheres, for which r is measured from centre to centre. For a satellite, r must therefore be the distance from the Earth's centre, not the height above the surface - a classic one-mark trap. The inverse-square dependence means that doubling the distance reduces the force to one quarter.
该定律严格适用于质点；牛顿证明对均匀球体同样成立，此时 r 为两球心之间的距离。因此对卫星而言，r 必须取到地心的距离，而不是离地面的高度——这是经典的一分陷阱。平方反比意味着距离加倍，引力减为原来的四分之一。
9. Gravitational Field Strength and Field Lines | 引力场强度与引力场线
Gravitational field strength g at a point is defined as the gravitational force per unit mass acting on a small test mass placed at that point. It is a vector directed towards the centre of the attracting body. Combining the definition with the law of gravitation shows that g depends only on the mass of the body producing the field and the distance from its centre, not on the test mass itself.
某点的引力场强度 g 定义为：放在该点的微小检验质量所受引力与其质量的比值。它是矢量，方向指向施力天体的中心。把该定义与万有引力定律结合，可看出 g 只取决于产生引力场的天体质量以及到其中心的距离，与检验质量本身无关。
g = F / m = GM / r² （球体外部）
Outside a uniform sphere, g follows an inverse-square law and the field lines point radially inwards with increasing spacing. Inside a uniform sphere, only the mass enclosed by the sphere of radius r contributes, and the field strength falls linearly with r, reaching zero at the centre. Field lines are always drawn with arrows pointing towards the mass and never cross one another.
在均匀球体外部，g 遵循平方反比规律，引力场线沿径向指向球心且间距越来越疏。在均匀球体内部，只有半径 r 所包围的那部分质量起作用，场强随 r 线性减小，在球心处为零。画场线时必须带指向天体的箭头，且任意两条场线不能相交。
球体内部：g = (4/3)πGρr （∝ r）
10. Gravitational Potential and Potential Energy | 引力势与引力势能
Gravitational potential V at a point is the work done per unit mass in bringing a small test mass from infinity to that point. Because the gravitational force is attractive, the field does positive work as the mass falls inwards, so the potential is negative and increases (becomes less negative) with distance, reaching zero at infinity. Gravitational potential energy is the corresponding quantity for a mass m.
某点的引力势 V 定义为：把微小检验质量从无穷远处移到该点过程中，引力所做的功与检验质量的比值。由于引力是吸引力，质量向内移动时引力做正功，因此引力势为负值，且随距离增大而增大（负得越少），在无穷远处趋于零。引力势能则是质量为 m 的物体对应的能量量。
V = − GM / r Eₚ = mV = − GMm / r ΔW = mΔV
The negative sign is not decorative: it encodes the fact that a bound system has less energy than a free one, and that energy must be supplied to move a satellite to a higher orbit. Note also that V is a scalar, so potentials from several masses are added algebraically. Escape speed follows directly by equating kinetic energy to the magnitude of the potential energy at the surface.
负号绝非装饰：它体现了束缚态系统的能量低于自由状态，也说明把卫星送到更高轨道必须输入能量。另外 V 是标量，多个天体产生的引力势可直接代数相加。逃逸速率可由动能等于表面处势能大小直接求出。
vesc = √2GM / R = √2gR
11. Orbits, Kepler's Third Law and Geostationary Satellites | 轨道、开普勒第三定律与地球同步卫星
and substituting v = 2πr / T gives the relationship between orbital radius and period. This is Kepler' cube of the orbital radius, and the constant depends only on the mass of the central body.
对做圆轨道运动的卫星而言，万有引力即为向心力。令 GMm / r² 等于 mv² / r 可得轨道速率；再代入 v = 2πr / T 可得轨道半径与周期的关系。这就是开普勒第三定律：周期的平方与轨道半径的立方成正比，比例常量只取决于中心天体的质量。
v = √GM / r T² = (4π² / GM) r³
ly 24 hours, it orbits above the equator, and it orbits in the same direction as the Earth' bit satellite, by contrast, has a much shorter period and can scan the whole surface as the Earth turns beneath it.
地球同步卫星必须满足三个条件：周期恰为 24 小时、轨道位于赤道正上方、运行方向与地球自转方向相同。由此可得轨道半径约 4.2 × 10⁷ m，即距地面约 3.6 × 10⁷ m。相比之下，极轨卫星周期短得多，能随着地球自转扫描整个地表。
Astronauts on the International Space Station feel "weightless" not because gravity is absent but because they and the station are in free fall with the same acceleration; the normal contact force on them is zero. Their mass is unchanged, and g at that altitude is still close to 9 m s⁻².
国际空间站上的宇航员感到"失重"，并非因为那里没有引力，而是因为人与空间站以相同加速度自由下落，作用在人体上的支持力为零。他们的质量并未改变，该高度处的 g 依然接近 9 m s⁻²。
12. Exam Technique and Common Errors | 应试技巧与常见失分点
Most marks are lost through a handful of recurring mistakes rather than through a lack of understanding. Keep a mental checklist: convert all angular rates to rad s⁻¹, use centre-to-centre distances for r, resolve forces rather than guessing which force is "the centripetal force", and always state the direction of the resultant force. When a question mentions "minimum speed", write down the condition that the contact force or tension is zero before substituting numbers.
多数失分并非源于理解不足，而是几个反复出现的错误。建议建立一张清单：把角速度统一换算为 rad s⁻¹；r 一律取两球心距离；用受力分解而不是凭感觉指定"哪个力是向心力"；并明确写出合力的方向。遇到"最小速率"时，先写出支持力或张力为零的临界条件，再代入数值计算。
| Common error | 常见错误 | Correct approach | 正确处理 |
| Using r = height above the surface | 把 r 取成离地高度 | r = R + h, measured centre to centre | r = R + h，取球心距 |
| Drawing an extra centripetal force arrow | 额外画出"向心力"箭头 | Label only real forces, then resolve | 只标真实力，再做分解 |
| Taking gravitational potential as positive | 把引力势取为正值 | V = − GM / r, zero at infinity | V = − GM / r，无穷远处为零 |
| Ignoring that v changes in a vertical circle | 忽略竖直圆周中 v 的变化 | Use energy conservation between points | 用机械能守恒连接各点 |
| Quoting "centrifugal force" as real | 把"离心力"当成真实力 | Use centripetal resultant force | 用指向圆心的合力解释
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