📚 A Level Physics Circular Motion, Centripetal Force and Gravitation: Key Exam Points | A Level 物理圆周运动、向心力与万有引力考点全解析
/r and F = GMm/r² - reappear in questions about cars turning corners, satellites in orbit, planets around the Sun and astronauts apparently floating in a space station. Because the physics is so compact, examiners test understanding rather than memory: they want to see that you can identify which real force provides the centripetal resultant, and that you can move confidently between linear and angular quantities.
圆周运动与万有引力是 A Level 物理中考查密度最高的板块之一。同样的几个公式——v = rω、a = v²/r、F = mv²/r 与 F = GMm/r²——会反复出现在汽车转弯、卫星轨道、行星绕日以及空间站中宇航员看似悬浮的题目里。正因为物理内容非常集中，考官考的往往是理解而不是记忆：他们要看到你能判断出究竟是哪一个真实力提供了向心合力，并且能在线量与角量之间自如切换。
1. Describing Circular Motion: The Basic Quantities | 圆周运动的基本描述量
volution is 2π rad. Angular velocity ω is the rate of change of angular displacement, ω = Δθ/Δt, measured in rad s⁻¹. The period T is the time taken for one complete revolution, and the frequency f is the number of revolutions per second, so f = 1/T and T = 2π/ω. The linear speed is the arc length travelled per unit time, v = 2πr/T, which links directly to angular velocity through v = rω.
在圆周运动中，物体沿半径为 r 的固定圆轨迹运动。角位移 Δθ 以弧度为单位，一整圈对应 2π rad。角速度 ω 是角位移的变化率，ω = Δθ/Δt，单位为 rad s⁻¹。周期 T 是完成一整圈所需的时间，频率 f 是每秒转过的圈数，因此 f = 1/T，T = 2π/ω。线速度是单位时间内走过的弧长，v = 2πr/T，它与角速度的关系为 v = rω。
ω = Δθ / Δt = 2π / T = 2πf v = 2πr / T = rω a = v² / r = rω²
A crucial point follows immediately: in uniform circular motion the speed is constant but the velocity is not, because the direction of motion changes continuously. Any changing velocity means acceleration, and that is exactly where the next section begins.
由此立刻得到一个关键结论：在匀速圆周运动中，速率不变但速度在变，因为运动方向持续改变。速度发生变化就意味着存在加速度，这正是下一节讨论的起点。
- Angle in radians: θ = arc length / radius, so s = rθ.
弧度定义：θ = 弧长 / 半径，即 s = rθ。 - Always convert rev min⁻¹ to rad s⁻¹ before substituting: multiply by 2π/60.
代入公式前务必把 rev min⁻¹ 换算成 rad s⁻¹：乘以 2π/60。 - Angular velocity is the same for all points on a rigid rotating body, but linear speed depends on r.
刚体转动时各点角速度相同，但线速度取决于 r。
2. Centripetal Acceleration: Direction and Magnitude | 向心加速度的方向与大小
tant. This acceleration is called centripetal acceleration and it always points towards the centre of the circle. Its magnitude is a = v²/r, which can also be written as a = rω² or a = vω. The direction is perpendicular to the instantaneous velocity at every moment, so the acceleration changes direction as the object moves round, always pointing at the centre.
由于速度矢量的方向不断改变，做匀速圆周运动的物体虽然速率不变，却仍然有加速度。这个加速度叫做向心加速度，方向始终指向圆心。其大小为 a = v²/r，也可写成 a = rω² 或 a = vω。它的方向在每一时刻都垂直于瞬时速度，因此随着物体绕行，加速度方向不断改变，但始终指向圆心。
roximately vΔθ in magnitude and points towards the centre. Dividing by the small time Δt gives a = v(Δθ/Δt) = vω = v²/r, using ω = v/r. Knowing this derivation is worth marks in written papers.
你可以用矢量图推导 a = v²/r。取两个长度均为 v 的速度矢量，夹角为小角度 Δθ。速度变化量 Δv 的大小约为 vΔθ，方向指向圆心。除以很小的时间 Δt，得到 a = v(Δθ/Δt) = vω = v²/r，其中用到 ω = v/r。掌握这个推导过程在笔试中是可以得分的。
- a is towards the centre; v is tangential. They are perpendicular, never parallel.
a 指向圆心；v 沿切线方向。两者垂直，绝不平行。 - For a fixed r, doubling the speed quadruples the acceleration.
r 固定时，速率加倍，加速度变为原来的四倍。 - On a rotating disc, points further from the centre have greater a because a = rω².
在转盘上，离圆心越远的点加速度越大，因为 a = rω²。
3. Centripetal Force Is a Resultant, Not a New Force | 向心力是合力，不是一种新的力
This is the single most important idea in the topic. Centripetal force is not an extra force to be drawn on a free-body diagram. It is the name we give to the resultant force directed towards the centre. In every problem you must first identify the real forces acting - weight, normal contact force, tension, friction, gravitational attraction, electrostatic force - then resolve them radially. That radial resultant equals mv²/r.
这是本专题最重要的一条观念。向心力不是在受力分析图上额外加上的一个力，而是我们对指向圆心的合力所起的名字。每道题都必须先找出真实存在的力——重力、支持力、张力、摩擦力、万有引力、静电力——再把它们沿半径方向分解。这个径向合力等于 mv²/r。
F = m v² / r = m r ω² = m v ω
Because the centripetal force is always perpendicular to the velocity, it does no work on the object. The kinetic energy of a body in uniform circular motion therefore stays constant, and the force changes only the direction of motion, never the speed. This is a favourite one-mark explanation question.
由于向心力始终垂直于速度，它对物体不做功。因此做匀速圆周运动的物体动能保持不变，这个力的作用只是改变运动方向，而不改变速率。这是一个很常见的一分解释题。
- Never write "centripetal force" as a separate arrow alongside weight and tension.
千万不要把向心力与重力、张力并列为一条独立的箭头。 - Centripetal force is a resultant; "centrifugal force" is not a real force in an inertial frame.
向心力是合力；在惯性系中离心力不是一个真实的力。 - If the radial resultant disappears, the object flies off along the tangent, not radially outward.
若径向合力消失，物体沿切线飞出，而不是沿半径向外飞。
4. Horizontal Turns, Banked Tracks and the Conical Pendulum | 水平转弯、倾斜弯道与圆锥摆
qual to the required centripetal force gives μmg = mv²/r, so the maximum safe speed is v = √μgr no friction needed, the horizontal component of the normal contact force provides the centripetal force and tanθ = v²/(rg).
汽车在平直路面上转弯时，依靠摩擦力提供向心力。令摩擦力等于所需向心力可得 μmg = mv²/r，因此最大安全速率为 v = √μgr。注意质量被约掉了——在轮胎与路面条件相同时，更重的车也能以同样速率转弯。如果路面以角度 θ 倾斜且不需要摩擦力，则由支持力的水平分量提供向心力，tanθ = v²/(rg)。
ing horizontally towards the centre gives T sinθ = mω²r with r = L sinθ. Combining these leads to cosθ = g/(ω²L), which sets the angle for a given angular velocity.
圆锥摆中，系在绳上的小球沿水平圆运动，绳扫出一个圆锥面。竖直方向分解得 T cosθ = mg，水平指向圆心方向分解得 T sinθ = mω²r，其中 r = L sinθ。联立可得 cosθ = g/(ω²L)，它决定了给定角速度下的张角。
- Flat corner: friction supplies F; condition is μmg ≥ mv²/r.
平直弯道：摩擦力提供 F；条件为 μmg ≥ mv²/r。 - Banked corner: tanθ = v²/(rg), independent of mass.
倾斜弯道：tanθ = v²/(rg)，与质量无关。 - Conical pendulum: combine T cosθ = mg with T sinθ = mω²r.
圆锥摆：联立 T cosθ = mg 与 T sinθ = mω²r。
5. Vertical Circular Motion: Top, Bottom and Critical Speed | 竖直圆周运动：顶部、底部与临界速率
int both the weight and the contact force (or tension) point towards the centre, so mg + N = mv²/r. The minimum speed for the object to stay on the circle occurs when the contact force just falls to zero: mg = mv²/r, giving vmin = √gr at the top.
在竖直圆周运动中，重力始终向下，因此它对径向合力的贡献随位置而变。在最高点，重力与接触力（或张力）都指向圆心，故 mg + N = mv²/r。物体能维持圆周运动的最小速率出现在接触力恰好降为零时：mg = mv²/r，得最高点 vmin = √gr。
viest at the bottom of a loop. Applying conservation of energy between the bottom and the top, ½mv_b² = ½mv_t² + mg(2r), and substituting vt = √gr gives the minimum speed at the bottom, vb = √5gr.
在最低点，接触力向上而重力向下，因此径向合力为 N − mg = mv²/r。这就是在环形轨道底部感觉最重的原因。在底部与顶部之间应用能量守恒，½mv_b² = ½mv_t² + mg(2r)，代入 vt = √gr 可得底部的最小速率 vb = √5gr。
Top: mg + N = m v² / r → vmin = √gr Bottom: N − mg = m v² / r → vmin = √5gr
- ing or a bead on the inside of a track cannot push, so N = 0 gives the critical condition.
绳或轨道内侧的珠子不能推物体，因此 N = 0 就是临界条件。 - A rod or a tube can push outward, so the critical speed at the top is zero.
杆或管可以向外推，因此顶部临界速率为零。 - Maximum tension usually occurs at the bottom, not at the top.
最大张力通常出现在最低点，而不是最高点。
6. Common Errors and Mark-Losing Mistakes | 常见错误与失分点
Most lost marks in this topic come from a small set of repeated errors. The most frequent is treating centripetal force as an additional force and drawing it beside weight, tension or friction. The second is confusing the direction of velocity with the direction of acceleration. The third is forgetting that in vertical motion the weight must always be included in the radial equation, even when it points away from the centre at the bottom.
本专题的失分大多来自少数几个反复出现的错误。最常见的是把向心力当成一个额外的力，与重力、张力或摩擦力并列画出。其次是混淆速度方向与加速度方向。第三是在竖直圆周运动中忘记重力必须始终出现在径向方程里，即使它在最低点是背离圆心的。
estion gives a period in hours, such as 24 h for a geostationary satellite, converting to 8.64 × 10⁴ s is part of the answer, not an optional extra. Finally, when a question asks for the source of the centripetal force, weak answers say "the force pulling it in" - strong answers name the actual force.
单位错误同样代价高昂。速率必须用 m s⁻¹，半径用 m，角速度用 rad s⁻¹，周期用秒，而不是分或小时。当题目给出以小时为单位的周期时，比如同步卫星的 24 h，换算成 8.64 × 10⁴ s 本身就是得分点，而不是可选项。最后，当题目问到向心力的来源时，弱答案会写"把它拉向里的那个力"，而强答案会明确指出究竟是哪一个力。
- Do not add "centripetal force" as a separate arrow on free-body diagrams.
不要在受力图上把"向心力"作为独立箭头加上去。 - Do not mix minutes, hours or revolutions per minute with SI units.
不要把分钟、小时或每分钟转数与 SI 单位混用。 - State the named force: friction, tension, gravity, normal contact force, drag.
要写出具体的力：摩擦力、张力、重力、支持力、阻力。 - Check whether the question needs force, acceleration or speed - they are different answers.
看清题目要的是力、加速度还是速率——三者答案不同。
7. Newton's Law of Universal Gravitation | 牛顿万有引力定律
quare of the distance between their centres. This is written F = GMm/r², where G = 6.67 × 10⁻¹¹ N m² kg⁻² is the gravitational constant. The force acts along the line joining the centres, and by Newton's third law the two bodies pull on each other with equal and opposite forces.
任意两个质点之间的引力与它们质量的乘积成正比，与它们中心之间距离的平方成反比，写作 F = GMm/r²，其中 G = 6.67 × 10⁻¹¹ N m² kg⁻² 是引力常量。引力沿两球心连线作用，并且根据牛顿第三定律，两个物体之间的相互吸引力大小相等、方向相反。
The inverse-square dependence is the key feature to comment on in exam answers. If the separation doubles, the force falls to one quarter. For a spherically symmetric body such as a planet, the mass can be treated as concentrated at its centre, which is why r is measured from the centre of the planet and not from its surface. The distance r in orbital problems is therefore planetary radius plus orbital altitude.
平方反比关系是简答题中最需要点明的特征。距离加倍，引力变为原来的四分之一。对于行星这类球对称天体，可以把质量看作集中在球心，这就是为什么 r 要从行星中心而不是表面量起。因此在轨道问题中，r 等于行星半径加上轨道高度。
F = G M m / r² G = 6.67 × 10⁻¹¹ N m² kg⁻²
8. Gravitational Field Strength and the Value of g | 引力场强度与 g 的取值
d in N kg⁻¹. Substituting the law of gravitation gives g = GM/r² for a point or a spherically symmetric mass, where r is measured from the centre. At the surface of a planet of radius R, this becomes g = GM/R².
某点的引力场强度定义为放在该点的小试探质量所受的引力与其质量之比，g = F/m，单位为 N kg⁻¹。代入万有引力定律，对于质点或球对称质量体可得 g = GM/r²，其中 r 从中心量起。在半径为 R 的行星表面，则为 g = GM/R²。
This equation explains several familiar facts at once. Field strength decreases with height above the surface, following an inverse-square law. Because the Moon has a smaller mass and a smaller radius than the Earth, its surface g is only about 1.6 N kg⁻¹. On Earth, GM/R² gives approximately 9.81 N kg⁻¹, which is why g is often quoted as 9.8 m s⁻²: the two units are numerically identical because force per unit mass has the same dimensions as acceleration.
这个式子可以一次解释好几个熟悉的现象。场强随离地高度增加而减小，遵循平方反比规律。由于月球的质量和半径都比地球小，其表面 g 仅为约 1.6 N kg⁻¹。在地球上，GM/R² 给出约 9.81 N kg⁻¹，这也是 g 常被写作 9.8 m s⁻² 的原因：两个单位在数值上相同，因为单位质量所受的力与加速度具有相同的量纲。
- g is a vector: its direction is towards the centre of the mass.
g 是矢量：方向指向质量体的中心。 - Inside a uniform sphere, g decreases linearly with r and is zero at the centre.
在均匀球体内部，g 随 r 线性减小，在球心处为零。 - Satellites in orbit still experience gravity; that is what keeps them in orbit.
轨道上的卫星仍然受到引力作用，正是引力维持了它的轨道。
9. Satellites and Kepler's Third Law | 卫星运动与开普勒第三定律
nal attraction is the centripetal force. Setting GMm/r² = mv²/r and cancelling m gives v = √GM/r. The orbital speed is therefore independent of the satellite's mass and depends only on the mass of the central body and the orbital radius. A larger orbit means a slower satellite, which is a common exam comment.
对于做圆轨道运动的卫星，万有引力就是向心力。令 GMm/r² = mv²/r 并约去 m，得 v = √GM/r。因此轨道速率与卫星自身质量无关，只取决于中心天体的质量和轨道半径。轨道越大，卫星越慢，这是简答题中常见的结论。
Substituting v = 2πr/T into that result and rearranging gives T² = (4π²/GM
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