📚 The Origins of Centripetal Forces | 向心力的来源
In circular motion an object keeps changing direction, so its velocity vector changes even when its speed is constant. That change demands an acceleration directed towards the centre of the circle, and the resultant force that produces this centre-seeking acceleration is called the centripetal force. The crucial point for A-Level Physics is that this force is never a mysterious extra push: it is always supplied by identifiable real forces such as tension, friction, gravity or a contact force. This article explains where centripetal forces come from and how to analyse them in CIE A-Level Physics.
在圆周运动中，物体不断改变方向，因此即使速率不变，它的速度矢量也在变化。这种变化需要一个指向圆心的加速度，而产生这个向心加速度的合力就叫做向心力。对 A-Level 物理而言，关键的一点是：向心力绝不是某种神秘的额外推力，它总是由可识别的真实力提供，例如张力、摩擦力、万有引力或接触力。本文将解释向心力的来源，以及如何在 CIE A-Level 物理中分析它们。
1. Circular Motion and a Velocity That Keeps Turning | 圆周运动与不断转向的速度
dometer and still be accelerating, because its direction of travel is changing at every instant. Acceleration is defined as the rate of change of velocity, a = Δv/Δt, and Δv here is a vector subtraction, not a simple arithmetic subtraction of two numbers.
速率说明物体运动有多快，而速度还包含方向。环岛里的一辆车速度表可以稳定显示 20 m s⁻¹ 却仍在加速，因为它的运动方向每一瞬间都在改变。加速度定义为速度的变化率 a = Δv/Δt，而这里的 Δv 是矢量相减，不是两个数值的简单代数相减。
on takes the period T = 2π/ω = 2πr/v, and the frequency is f = 1/T. Exam questions very often give T, f or ω instead of v, so converting between these quantities is the first skill to master.
对于以恒定速率 v 在半径 r 的圆周上运动的物体，每秒扫过的角度就是角速度 ω，且 v = ωr。完成一整圈所需的时间是周期 T = 2π/ω = 2πr/v，频率 f = 1/T。考题经常给出 T、f 或 ω 而不是 v，因此在这些量之间换算，是首先要掌握的能力。
- v - speed along the tangent, in m s⁻¹ | v - 沿切线方向的速率，单位 m s⁻¹
ω - angular velocity in rad s⁻¹, where v = ωr | ω - 角速度，单位 rad s⁻¹，且 v = ωr
T - period of one revolution in s | T - 转一圈的周期，单位 s
f - frequency in Hz, f = 1/T | f - 频率，单位 Hz，f = 1/T
2. What a Centripetal Force Really Is | 向心力究竟是什么
given by F = mv²/r, where m is the mass, v is the speed and r is the radius. It is defined by the job it does, not by its origin: "centripetal" simply means "centre-seeking".
向心力是作用在做圆周运动的物体上的合力，方向指向圆心。它的大小由 F = mv²/r 给出，其中 m 是质量，v 是速率，r 是半径。它是按"所起的作用"来定义的，而不是按"来源"来定义的：centripetal 的意思就是"指向圆心的"。
ational attraction, the normal contact force from a track, the electric force and the magnetic force can each play this role, and sometimes two of them act together. Newton' cond law is simply applied along the radius: the radial resultant equals ma with a = v²/r.
这就是为什么它不是"运动变成圆周运动时才出现"的一种新力。绳中的张力、轮胎与路面之间的摩擦力、万有引力、轨道提供的法向接触力、电场力与磁场力，都可以扮演这个角色，有时其中两种力同时起作用。牛顿第二定律只是沿着半径方向应用：径向合力等于 ma，其中 a = v²/r。
Examiners therefore insist that you never draw an extra arrow labelled "centripetal force" alongside the real forces on a free-body diagram. If you do, you double count and the equation will not balance. Draw only the real forces - weight, normal contact, tension, friction, drag - then resolve along the radius.
因此考官反复强调：在受力图上，绝不要在真实力之外再画一个标注为"向心力"的箭头。这样做会重复计算，方程就不平衡了。只画真实的力——重力、法向接触力、张力、摩擦力、阻力——然后沿半径方向分解。
3. The Working Equations and Their Units | 常用公式及其单位
Every circular-motion calculation in CIE A-Level Physics comes from one acceleration and one force equation, written in whichever form matches the data you are given. Learn all three forms of each, because choosing the right form saves minutes in the exam.
CIE A-Level 物理中所有圆周运动计算都来自一个加速度公式和一个力公式，只需根据题目给出的数据选择合适的形式。把每个公式的三种形式都记牢，考场上选对形式能省下好几分钟。
a = v²/r = ω²r = 4π²r/T²
F = mv²/r = mω²r = 4π²mr/T²
mω²r, since ω is in rad s⁻¹ and the radian is dimensionless. If your final unit is not N, an algebraic slip has crept in.
每次都要检查单位。牛顿等于 kg m s⁻²，因此 mv²/r 给出 kg × (m s⁻¹)² ÷ m = kg m s⁻²。对 mω²r 也同样成立，因为 ω 的单位 rad s⁻¹ 是无量纲的。如果最终单位不是 N，说明代数上出了错。
- Use v²/r when the question gives speed or the radius and period are not needed. | 当题目给出速率，或不需要半径与周期时，用 v²/r。
Use ω²r when the question gives angular velocity or a rate of rotation in rev min⁻¹. | 当题目给出角速度，或转速以 rev min⁻¹ 给出时，用 ω²r。
Use 4π²r/T² when the question gives a period or a time for one orbit. | 当题目给出周期或绕行一圈的时间时，用 4π²r/T²。
4. Why the Force Must Point to the Centre | 为什么力必须指向圆心
ions, so their difference Δv points inwards, roughly towards the centre, and its magnitude is approximately vΔθ. Dividing by Δt gives Δv/Δt ≈ v(Δθ/Δt) = vω = v²/r, directed towards the centre in the limit Δt → 0.
取一段很短的时间间隔 Δt，物体转过一个小角度。前后两个速度矢量长度相同、方向不同，因此它们的差 Δv 指向内侧、大致指向圆心，大小约为 vΔθ。除以 Δt 得 Δv/Δt ≈ v(Δθ/Δt) = vω = v²/r，当 Δt → 0 时方向严格指向圆心。
Because the force is always perpendicular to the velocity, it does no work on the body. There is no component of the centripetal force along the direction of motion, so the kinetic energy and the speed stay constant while the direction of motion keeps turning. This is a favourite one-mark explanation in exam papers.
由于这个力始终垂直于速度，它对物体不做功。向心力沿运动方向没有分量，因此动能与速率保持不变，只有运动方向不断改变。这是考卷中很受欢迎的一分简答题。
5. Source 1: Tension in a String or Cable | 来源一：绳或缆中的张力
Whirl a rubber bung on a nylon thread and the tension in the thread is the centripetal force. The tension pulls the bung inwards; the bung pulls the thread outwards, and if the tension needed exceeds the breaking stress of the thread, the thread snaps and the bung flies off along the tangent. Hammer throw and discus events use exactly the same physics.
用尼龙线甩一个橡皮塞时，线中的张力就是向心力。张力把橡皮塞拉向圆心；橡皮塞也把线向外拉，一旦所需的张力超过线的抗拉极限，线就会断裂，橡皮塞沿切线方向飞出。掷链球和掷铁饼用的正是同一套物理。
es T cos θ = mg and horizontally T sin θ = mv²/r with r = L sin θ. Dividing the two equations eliminates the tension and gives the key result.
在锥摆中，摆球系在一根长 L 的绳上，在水平面内做圆周运动，绳与竖直方向成 θ 角。竖直方向分解得 T cos θ = mg，水平方向 T sin θ = mv²/r，其中 r = L sin θ。两式相除消去张力，得到关键结果。
tan θ = v²/(rg) T = 2π√L cos θ / g
eak there. At the top, T + mg = mv²/r, and the minimum speed for the string to stay taut is found by setting T = 0.
在竖直平面内用绳做圆周运动时，张力沿圆周各处不同。在最低点 T − mg = mv²/r，此处绳最容易被拉断。在最高点 T + mg = mv²/r，令 T = 0 即可求出绳子保持绷紧的最小速率。
vmin at the top of a vertical circle = √gr
At the minimum speed the weight alone provides exactly the required centripetal force, and the string just goes slack any slower than that.
在最小速率下，仅靠重力就恰好提供所需的向心力；一旦比这个速率更慢，绳就会松弛。
6. Source 2: Friction on Roads, Tracks and Turntables | 来源二：路面、赛道与转盘上的摩擦力
ere is no sliding. The required value is f = mv²/r, and it cannot exceed the limiting value μmg, so the maximum safe speed follows. Notice that the mass cancels, so a loaded lorry and a small car can corner at the same speed if the tyres and road are identical.
对水平路面上的汽车来说，唯一可用的水平力是轮胎与路面之间的摩擦力，而且因为没有相对滑动，它是静摩擦力。所需值为 f = mv²/r，最大不能超过极限值 μmg，于是得到最大安全速率。注意质量被消去，因此只要轮胎与路面条件相同，满载卡车和小汽车能以同样的速率过弯。
f = mv²/r ≤ μmg → vmax = √μgr
h no friction, the horizontal component of the normal contact force supplies everything, giving tan θ = v²/(rg), which is the design speed of the bend.
同样的推理可以解释转盘上的硬币、弯道上的列车以及倾斜过弯的自行车。在无摩擦的倾斜赛道上，法向接触力的水平分量提供全部向心力，得 tan θ = v²/(rg)，这就是弯道的设计速率。
Do not confuse this with the friction that opposes forward motion or with braking friction. The lateral friction that steers a car round a bend acts perpendicular to the car's velocity, so it does no work either; a car that takes a bend at constant speed uses grip, not engine power, to change direction.
不要把它与阻碍前进的摩擦力或刹车摩擦力混为一谈。使汽车转弯的侧向摩擦力垂直于车的速度，因此同样不做功；一辆以恒定速率过弯的车，改变方向靠的是抓地力，而不是发动机功率。
7. Source 3: Gravitational Attraction in Orbits | 来源三：轨道运动中的万有引力
For a satellite, the Moon around the Earth or a planet around the Sun, the centripetal force is the gravitational force. Setting Newton's law of gravitation equal to the circular-motion requirement is the standard starting point for every orbital question.
对于人造卫星、绕地球运动的月球或绕太阳运动的行星，向心力就是万有引力。把牛顿万有引力定律与圆周运动的要求等式相等，是每一道轨道题的标准起点。
GMm/r² = mv²/r → v = √GM/r
Squaring the orbital speed and substituting v = 2πr/T gives Kepler' rth: only at that radius does T equal 24 hours while the satellite stays above the same point on the equator.
把轨道速率平方并代入 v = 2πr/T，得到开普勒第三定律的形式 T² = 4π²r³/(GM)。这一关系解释了为什么地球静止轨道卫星必须位于距地心约 3.6 × 10⁷ m 处：只有在这个半径上，T 才等于 24 小时，卫星才能停在地球赤道上空同一点。
Astronauts in orbit feel weightless not because gravity has vanished but because they and the spacecraft are falling towards the Earth together, so the normal contact force on them is zero. Gravity is still providing the centripetal force that keeps them in orbit - a very common exam explanation.
轨道上的宇航员感觉失重，不是因为重力消失了，而是因为他们与飞船一起朝地球自由下落，所以他们受到的法向接触力为零。重力仍在提供使他们留在轨道上的向心力——这是考卷上非常常见的解释题。
8. Source 4: Normal Contact Forces from Tracks and Surfaces | 来源四：轨道与表面提供的法向接触力
A ball rolling on the inside of a loop, a marble in a curved tube or a toy car on a curved track is held in its circle by the normal contact force from the surface. At the top of the loop the contact force and the weight both point towards the centre, so they add.
小球沿竖直圆环内侧滚动、弹珠在弯曲管道中运动或玩具车在弯曲轨道上行驶时，把它约束在圆周上的是表面提供的法向接触力。在圆环最高点，接触力与重力都指向圆心，因此两者相加。
At the top of a loop: N + mg = mv²/r
Setting N = 0 gives the minimum speed for a ball to keep contact with the track, v = √gr, which depends only on the radius and not on the mass. At the bottom of the loop the contact force must support the
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