📚 Describing circular motion | 描述圆周运动
Circular motion is the movement of an object along the circumference of a circle or a circular path. It is a fundamental topic in A-Level Physics, covering concepts such as angular displacement, angular velocity, centripetal acceleration, and centripetal force. Understanding how to describe circular motion enables us to analyse everything from a car round a bend to the orbit of a planet.
圆周运动是物体沿圆周或圆形路径移动的运动形式。这是 A-Level 物理的基础课题，涵盖了角位移、角速度、向心加速度和向心力等概念。理解如何描述圆周运动，可以帮助我们分析从汽车转弯到行星轨道等各种现象。
1. Introduction to Circular Motion | 圆周运动入门
An object is said to be in circular motion when its trajectory is a circle of constant radius. In uniform circular motion, the speed of the object remains constant, but its velocity changes because the direction of motion is continuously changing. The net acceleration is directed towards the centre of the circle, and a net inward force must act to maintain the circular path.
当物体的轨迹是一个半径不变的圆时，我们就说它在做圆周运动。在匀速圆周运动中，物体的速率保持不变，但由于运动方向不断改变，速度矢量是变化的。合加速度指向圆心，必须有指向圆心的净力来维持圆形的运动轨迹。
2. Angular Displacement and Radians | 角位移与弧度
ns, where one radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. Thus, θ = s / r, with s being the arc length. One full revolution corresponds to 2π radians (360°).
角位移 θ 衡量的是物体沿圆运动时半径线所扫过的角度，单位是弧度。1 弧度的定义是：弧长等于半径时，弧所对的圆心角的大小。因此，θ = s / r，其中 s 是弧长。完整一圈对应于 2π 弧度（360°）。
3. Angular Velocity and Linear Speed | 角速度与线速度
) speed v is related to ω by v = r ω, where r is the radius. This follows directly because the arc length per unit time is r × (angle swept per unit time).
角速度 ω 是角位移的变化率。对于匀速圆周运动，ω = Δθ / Δt，单位是 rad s⁻¹。线速度（切向速率）v 与 ω 之间的关系是 v = r ω，其中 r 为半径。这可以直接导出，因为单位时间内的弧长就是半径乘以单位时间内扫过的角度。
v = r ω
The direction of the angular velocity vector is given by the right-hand rule: if the fingers of the right hand curl in the direction of rotation, the thumb points along the axis of rotation.
角速度矢量的方向由右手定则确定：让右手手指弯曲指向旋转方向，则拇指所指的方向就是角速度的方向（沿转轴）。
4. Period and Frequency | 周期与频率
adians, angular velocity can also be expressed as ω = 2π / T = 2π f. The linear speed becomes v = 2π r / T.
周期 T 是完成一整圈所需的时间。频率 f 是单位时间内的圈数：f = 1 / T。由于一圈对应 2π 弧度，角速度也可以写成 ω = 2π / T = 2π f。线速度则表示为 v = 2π r / T。
ω = 2π f and v = 2π r / T
5. Centripetal Acceleration | 向心加速度
circle and is called centripetal acceleration, a. Its magnitude is given by a = v² / r = r ω². A simple derivation uses the change in velocity vectors over a small time interval to show the inward direction and the ratio of v² / r.
在匀速圆周运动中，尽管速率不变，但速度的方向在不断改变。加速度指向圆心，称为向心加速度 a。它的大小为 a = v² / r = r ω²。可以通过速度矢量在很短时间内隔内的变化量，简单推导出这一结果，并表明加速度方向向心。
a = v² / r = r ω²
Because this acceleration is perpendicular to the velocity, it does not change the speed, only the direction of motion.
因为向心加速度始终垂直于速度，它不改变速率，仅改变运动的方向。
6. Centripetal Force | 向心力
According to Newton' . It is not a new type of force, but rather a role played by a real force such as tension, friction, gravity, or the normal reaction, directed towards the centre.
根据牛顿第二定律，必须有净力来产生向心加速度。这个指向圆心的净力就叫做向心力：F = m a = m v² / r = m r ω²。向心力并不是一种新的力，而是由真实的力（例如张力、摩擦力、重力或法向反作用力）充当的角色，方向指向圆心。
F = m v² / r = m r ω²
If the force providing centripetal action suddenly disappears, the object will move off tangentially in a straight line.
如果提供向心作用的力突然消失，物体将沿切线方向做匀速直线运动。
7. Examples of Horizontal Circular Motion | 水平圆周运动实例
A common scenario is a car turning on a flat, horizontal road. The centripetal force is supplied by the sideway friction between the tyres and the road. If the friction is insufficient, the car will skid outwards. For a ball whirled on a string in a horizontal circle, the tension in the string provides the centripetal force. In a conical pendulum, the horizontal component of the string's tension points towards the centre and equals m v² / r.
一个常见情景是汽车在水平路面上转弯。向心力由轮胎与路面之间的横向摩擦力提供。如果摩擦力不足，汽车就会向外侧滑。对于用绳子在水平面内旋转的小球，绳的张力提供了向心力。在锥摆中，绳子张力的水平分量指向圆心，并等于 m v² / r。
Other examples include a cyclist leaning into a bend to generate a horizontal component of the normal reaction, and a charged particle moving perpendicular to a uniform magnetic field, where the magnetic force acts as the centripetal force.
其他例子还包括：骑自行车的人在弯道倾斜车身，以便法向反作用力产生水平分量；带电粒子垂直进入匀强磁场时，磁力充当向心力。
8. Vertical Circular Motion | 垂直圆周运动
When an object moves in a vertical circle, its speed is not constant because gravitational potential energy changes. The centripetal force at any point is the net inward force towards the centre. At the top of the circle, both weight and tension (or normal force) act downwards and together provide the required centripetal force. At the bottom, the tension must be greater than the weight to supply the upward centripetal force.
当物体在竖直平面内做圆周运动时，由于重力势能的变化，其速率并不恒定。任意位置的向心力都是指向圆心的净力。在圆周的最高点，重力和张力（或法向力）都向下，共同提供所需的向心力。在最低点，张力必须大于重力，以提供向上的向心力。
Top: T + mg = m v² / r
Bottom: T − mg = m v² / r
These expressions are essential for problems involving buckets of water swung overhead or roller coaster loops.
在处理水桶头顶画圈或过山车回环等问题时，这些表达式非常关键。
9. Applications to Orbital Motion | 天体运动应用
al force to the centripetal force gives: G M m / r² = m v² / r. This simplifies to the orbital speed v = √G M / r. The period of the orbit is T = 2π r / v, leading to T² ∝ r³ (Kepler's third law).
对于绕地球做圆周运动的卫星，向心力由万有引力提供。令万有引力等于向心力：G M m / r² = m v² / r。简化后得到轨道速率 v = √G M / r。轨道周期 T = 2π r / v，由此得出 T² ∝ r³（开普勒第三定律）。
v = √G M / r
This shows that satellites in lower orbits travel faster and have shorter periods, a result consistent with geostationary and polar orbit applications.
这表明轨道越低的卫星运行速度越快、周期越短，这个结果与地球同步轨道和极地轨道的应用相一致。
10. Summary and Key Equations | 总结与关键公式
Circular motion can be fully described using angular and linear quantities. The key relationships to remember are:
圆周运动可以用角量与线量全面描述。需要记住的关键关系式有：
v = r ω
a = v² / r = r ω²
F = m v² / r = m r ω²
ω = 2π / T = 2π f
Always identify the force that acts towards the centre, and remember that the centripetal force is never an extra force but the resultant of physical forces.
始终要找出指向圆心的实际作用力，并记住向心力决不是额外多出来的力，而是真实力在指向圆心方向上的合力。
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