Circular Motion and Gravitation Comprehensive Review | 圆周运动与万有引力综合复习

📚 Circular Motion and Gravitation Comprehensive Review | 圆周运动与万有引力综合复习

Circular motion and gravitation form one of the most frequently examined areas in IB Physics, appearing in both Paper 1 and Paper 2 across multiple question formats. This comprehensive review consolidates all essential concepts aligned with the IB syllabus objectives, enabling you to apply equations with confidence and analyse physical scenarios systematically in examinations.

圆周运动与万有引力是 IB 物理中考察频率最高的内容之一，在 Paper 1 与 Paper 2 中均以多种题型反复出现。本综合复习整合了与 IB 教学大纲学习目标直接相关的全部核心概念，帮助你自信地应用公式，并在考试中有条不紊地分析各类物理情境。

1. Key Definitions and Units | 基本定义与单位

Circular motion describes the movement of an object along a circular path. The fundamental quantities include period T, defined as the time for one complete revolution measured in seconds, and frequency f, defined as the number of revolutions per second measured in hertz. The angular displacement θ is measured in radians, where one full revolution corresponds to 2π radians.

圆周运动描述物体沿圆形路径的运动。基本物理量包括周期 T，即完成一整圈所需的时间，单位为秒；频率 f，即每秒钟完成的圈数，单位为赫兹。角位移 θ 的单位为弧度，一整圈对应 2π 弧度。

The radian is the angle subtended at the centre of a circle by an arc whose arc length equals the radius. Because 2π radians corresponds to 360°, we can convert using the proportion θ(rad) = θ(°)×π/180°.

弧度的定义是：圆上长度等于半径的弧所对应的圆心角大小。由于 2π 弧度对应 360°，换算关系为 θ(弧度) = θ(角度)×π/180°。

The angular velocity ω is the rate of change of angular displacement, defined as ω = Δθ/Δt. For uniform circular motion, this quantity is constant and equals 2πf or 2π/T.

角速度 ω 是角位移的变化率，定义为 ω = Δθ/Δt。对于匀速圆周运动，角速度为常量，且满足 ω = 2πf 或 ω = 2π/T。

• Period T = 2π/ω (周期 T = 2π/ω)
• Frequency f = 1/T (频率 f = 1/T)
• Relationship: ω = 2πf = 2π/T (角速度关系式：ω = 2πf = 2π/T)
• Unit of ω: rad s⁻¹ (ω 的单位：rad s⁻¹)

2. Centripetal Acceleration and Force | 向心加速度与向心力

An object moving in uniform circular motion is continuously accelerating because its velocity direction changes constantly, even though its speed remains unchanged. This acceleration is directed radially inward toward the centre of the circle and is known as centripetal acceleration.

做匀速圆周运动的物体始终处于加速状态，因为其速度方向不断改变，尽管速率保持不变。这一加速度方向沿半径指向圆心，称为向心加速度。

The magnitude of centripetal acceleration can be expressed in three equivalent forms:

向心加速度的大小具有三种等价表达形式：

a = v²/r = ω²r = 4π²r/T²

According to Newton’s second law, the net force producing this acceleration is the centripetal force, given by:

根据牛顿第二定律，产生该加速度的合力称为向心力，其表达式为：

F = mv²/r = mω²r = mωv

It is crucial to recognise that the centripetal force is not an additional independent force. Rather, it is the name given to the resultant of existing forces that points toward the centre. In different physical situations, the centripetal force may be provided by tension in a string, friction between tyres and road, gravitational attraction, or the normal reaction from a surface.

必须明确一点：向心力并不是额外独立的力。它是所有指向圆心的已有作用力的合力的名称。在不同情境中，向心力可由绳中张力、轮胎与地面的摩擦力、万有引力或接触面的支持力等提供。

3. Angular Velocity and Period | 角速度与周期

The linear speed v of an object in circular motion relates to its angular velocity through v = ωr. Substituting ω = 2π/T gives v = 2πr/T, which expresses the linear speed in terms of the period. These equations allow convenient conversion between angular and linear descriptions of rotational motion.

圆周运动中物体的线速度 v 与角速度的关系为 v = ωr。代入 ω = 2π/T 后得到 v = 2πr/T，即用周期表示线速度。这些公式便于在线量与角量描述之间进行转换。

For uniform circular motion, the angular velocity is constant, but the linear velocity is not constant because its direction continuously changes. The banking of roads and the design of racing tracks use the relationship between centripetal force and angle of inclination to increase the maximum safe speed without relying solely on friction.

在匀速圆周运动中，角速度恒定，但线速度并不恒定，因为其方向持续改变。道路弯道的外侧抬高以及赛车赛道设计正是利用向心力与倾斜角之间的关系，在不过度依赖摩擦力的条件下提高安全最高速度。

When a car negotiates an unbanked curve, the frictional force supplies the centripetal force. The maximum speed is therefore limited by the condition μmg = mv²/r, which simplifies to v_max = √(μgr).

当汽车在水平弯道上转弯时，摩擦力提供向心力。因此最大车速受到条件 μmg = mv²/r 的限制，化简后得 v_max = √(μgr)。

• v = ωr = 2πr/T (线速度与角速度、周期的关系)
• a = v²/r = ω²r = 4π²r/T² (向心加速度的三种形式)
• For a banked curve with angle θ: tanθ = v²/rg (倾斜弯道角度条件：tanθ = v²/rg)

4. Vertical Circular Motion | 竖直圆周运动

In vertical circular motion, gravitational force contributes to the net centripetal force, and the object’s speed typically varies as it moves around the circle. At the top of the loop, both the tension T and weight mg act downward, so the centripetal force is T + mg. At the bottom, tension acts upward while weight acts downward, yielding T – mg.

在竖直圆周运动中，重力参与了向心力的合成，物体速率通常随位置而改变。在圆周最高点，张力 T 与重力 mg 均向下，因此所需向心力为 T + mg；在最低点，张力向上而重力向下，因此所需向心力为 T – mg。

For an object to just complete the loop, the minimum speed at the top occurs when the centripetal force equals the object’s weight alone, meaning tension is zero. Setting mg = mv²/r gives the critical speed v_min = √(gr).

物体恰好能完成整圈运动时，最高点的最小速率出现在向心力恰好完全由重力提供的临界状态，即张力为零。令 mg = mv²/r，可得临界速率 v_min = √

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