Circular Motion | 圆周运动

📚 Circular Motion | 圆周运动

Circular motion is a fundamental topic in AQA Physics, describing the motion of an object along a circular path. Mastering the concepts of angular displacement, angular velocity, centripetal acceleration, and centripetal force is essential for solving exam problems on banking, conical pendulums, and vertical circles. This article provides a comprehensive, bilingual breakdown of every key point to ensure you fully grasp the principles and common pitfalls.

圆周运动是 AQA 物理的基础课题，描述物体沿圆周轨迹的运动。掌握角位移、角速度、向心加速度和向心力等概念，对于解决倾斜弯道、圆锥摆和竖直圆周等考题至关重要。本文以中英双语全面梳理每个考点，助你透彻理解原理并避开常见错误。

1. Angular Displacement and Radian Measure | 角位移与弧度制

Angular displacement θ is the angle through which a point or line has been rotated in a specified sense about a specified axis. In circular motion, it is the angle swept by the radius vector from the centre to the object.

角位移 θ 是指一个点或线绕指定轴沿指定方向转过的角度。在圆周运动中，它是从圆心到物体的半径矢量所扫过的角度。

The SI unit of angular displacement is the radian (rad). One radian is defined as the angle subtended at the centre of a circle by an arc whose length is equal to the radius of the circle.

角位移的国际单位是弧度 (rad)。一弧度的定义是：当一段圆弧的长度等于该圆的半径时，该圆弧所对的圆心角的大小。

The relationship between arc length s, radius r, and angular displacement θ in radians is given by:

弧长 s、半径 r 与以弧度为单位的角位移 θ 之间的关系为：

θ = s / r

Since the circumference of a full circle is 2πr, a full revolution corresponds to an angular displacement of 2π radians, equivalent to 360°.

由于整圆的周长为 2πr，一整周对应的角位移为 2π 弧度，等于 360°。

2. Angular Velocity | 角速度

Angular velocity ω quantifies the rate of change of angular displacement. For uniform circular motion, the angular velocity is constant in magnitude.

角速度 ω 量化了角位移随时间变化的快慢。在匀速圆周运动中，角速度的大小恒定。

Average angular velocity is defined as ω = Δθ / Δt. The instantaneous angular velocity is the limit as Δt approaches zero, ω = dθ/dt. Its unit is rad s⁻¹.

平均角速度定义为 ω = Δθ / Δt。瞬时角速度是当 Δt 趋近于零时的极限，即 ω = dθ/dt。单位为 rad s⁻¹。

For one complete revolution, the time taken is the period T, and the angular displacement is 2π rad. Therefore:

对于一整周，所需时间为周期 T，角位移为 2π rad。因此：

ω = 2π / T

Since frequency f = 1/T, we can also write ω = 2πf. Angular velocity is sometimes called angular frequency.

由于频率 f = 1/T，我们也可以写成 ω = 2πf。角速度有时也称为角频率。

3. Relationship Between Linear and Angular Velocity | 线速度与角速度的关系

The linear speed v of an object moving in a circle of radius r is related to its angular velocity by a simple yet powerful equation.

在半径为 r 的圆周上运动的物体的线速度 v 与角速度之间存在一个简洁而重要的关系式。

From the definition of radian measure, s = rθ. Differentiating with respect to time gives ds/dt = r (dθ/dt). Since ds/dt is the linear speed v and dθ/dt is ω, we obtain:

由弧度制定义 s = rθ，对时间求导得 ds/dt = r (dθ/dt)。因为 ds/dt 为线速度 v，dθ/dt 为 ω，所以得到：

v = ω r

Note that while v and ω may be constant in uniform circular motion, the direction of the velocity vector is continuously changing, always tangent to the circle.

注意，尽管在匀速圆周运动中 v 和 ω 的大小不变，但速度矢量的方向在不断变化，始终沿圆周的切线方向。

4. Centripetal Acceleration | 向心加速度

An object in uniform circular motion experiences an acceleration directed towards the centre of the circle. This is called centripetal acceleration, and it arises from the continuous change in the direction of the velocity vector.

做匀速圆周运动的物体具有指向圆心的加速度，称为向心加速度，它源于速度矢量方向的持续改变。

The magnitude of centripetal acceleration a_c is given by two equivalent expressions:

向心加速度 a_c 的大小由以下两个等价的表达式给出：

a_c = v² / r

a_c = ω² r

Substituting v = ωr into a_c = v²/r yields a_c = (ωr)²/r = ω²r, confirming their equivalence. These equations apply to any object moving at constant speed in a circular path.

将 v = ωr 代入 a_c = v²/r 可得 a_c = (ωr)²/r = ω²r，证实了它们的等价性。这些公式适用于任何在圆形路径上匀速运动的物体。

Although the speed is constant, the acceleration is non-zero because the velocity vector changes direction. The centripetal acceleration is always perpendicular to the velocity and points radially inward.

尽管速率恒定，但加速度不为零，因为速度矢量方向在变化。向心加速度始终垂直于速度，沿径向指向圆心。

5. Centripetal Force | 向心力

According to Newton’s second law, a net force is required to cause an acceleration. The net force that produces centripetal acceleration is called the centripetal force.

根据牛顿第二定律，产生加速度需要净外力。产生向心加速度的净外力称为向心力。

Using F = ma, the magnitude of the centripetal force is:

利用 F = ma，向心力的大小为：

F = m v² / r

F = m ω² r

Centripetal force is always directed towards the centre of the circle. It is not a new type of force; it is the resultant of real forces such as tension, gravity, friction, or the normal reaction.

向心力始终指向圆心。它不是一种新型的力，而是由真实力（例如张力、重力、摩擦力或支持力）的合力提供。

A common misconception is the idea of a ‘centrifugal force’ pushing outward. In an inertial frame of reference, no such outward force acts on the object. The sensation of being thrown outward is due to inertia.

一个常见的误解是存在向外推的“离心力”。在惯性参考系中，并没有这样的向外力作用于物体。被向外甩的感觉是惯性的表现。

6. Sources of Centripetal Force in Common Scenarios | 常见情境中的向心力来源

Identifying the force providing the centripetal acceleration is a key exam skill. Below are typical examples:

识别提供向心加速度的力是一项关键的考试技能。以下是一些典型例子：

Scenario 情境	Force supplying F_c 提供向心力的力
Car turning on a flat road 汽车在水平路面转弯	Friction between tyres and road 轮胎与路面的摩擦力
Car on a banked track (no friction) 无摩擦倾斜弯道上的汽车	Horizontal component of the normal reaction 支持力的水平分量
Conical pendulum 圆锥摆	Horizontal component of tension in the string 绳子张力的水平分量
Planet orbiting a star 行星绕恒星运动	Gravitational force 万有引力
Electron orbiting a nucleus (Bohr model) 电子绕核运动（玻尔模型）	Electrostatic attraction 静电吸引力

When solving problems, always draw a free-body diagram and resolve forces along the radial direction. Set the net inward force equal to mv²/r or mω²r.

解题时，始终画出受力分析图，并沿径向分解力。将向内的合力设为等于 mv²/r 或 mω²r。

7. Vertical Circular Motion | 竖直面内的圆周运动

In vertical circular motion, the speed is not constant due to gravity. Energy considerations become important, and the centripetal force requirement still holds at every point.

在竖直面内的圆周运动中，由于重力的影响，速率并不恒定。能量因素变得重要，同时向心力的要求在每个点仍成立。

Consider an object attached to a string moving in a vertical circle. At the top of the circle, both tension T and weight mg act downward, providing the centripetal force:

考虑一个系在绳上的物体在竖直面内做圆周运动。在最高点，张力 T 和重力 mg 都向下，一起提供向心力：

T + mg = m v² / r

For the object to just complete the circle, the tension at the top can be zero (critical condition). The minimum speed at the top is given by mg = m v_min²/r, so:

要使物体刚好完成圆周运动，最高点的张力可为零（临界条件）。此时最高点的最小速率满足 mg = m v_min²/r，因此：

v_min = √(g r)

At the bottom of the circle, the tension is maximum because it must support the weight and provide the centripetal force:

在最低点，张力最大，因为它既要平衡重力，又要提供向心力：

T – mg = m v² / r

Energy conservation between the top and bottom gives a relationship between speeds: assuming zero of potential energy at the bottom, v_top² = v_bottom² – 4gr. This helps find tensions at various points.

利用最高点和最低点之间的能量守恒可建立速率关系：设最低点势能为零，则 v_top² = v_bottom² – 4gr。这有助于求出各点的张力。

8. Energy in Circular Motion | 圆周运动中的能量

For uniform circular motion in a horizontal plane, the kinetic energy (½mv²) is constant because speed is constant. However, in vertical circles, kinetic energy and gravitational potential energy interchange.

对于水平面内的匀速圆周运动，动能 (½mv²) 恒定，因为速率不变。然而在竖直圆周中，动能与重力势能相互转化。

If non-conservative forces (like friction or air resistance) are negligible, total mechanical energy is conserved:

若非保守力（如摩擦或空气阻力）可忽略，则总的机械能守恒：

½ m v₁² + m g h₁ = ½ m v₂² + m g h₂

Applying this to a mass on a string in a vertical circle, the speed at any height can be deduced, and the centripetal force equation can then be used to compute tensions.

将其应用于竖直圆周中绳上的物体，可推导任意高度的速率，然后再用向心力方程计算张力。

In many exam problems, you will be asked to find the minimum height from which a mass must be released so that it loops the loop. This requires equating initial potential energy to the kinetic energy at the top plus the potential energy there.

在许多考题中，会要求你找出物体为完成翻圈而必须释放的最小高度。这需要令初始势能等于最高点的动能及其势能之和。

9. Conical Pendulum and Banked Curves | 圆锥摆与倾斜弯道

A conical pendulum consists of a mass moving in a horizontal circle at the end of a string that traces a cone. The string tension T provides both the vertical component balancing weight and the horizontal radial component providing centripetal force.

圆锥摆由一个在水平面内做圆周运动的重物构成，绳子划出一个圆锥面。绳的张力 T 提供的竖直分量平衡重力，水平径向分量提供向心力。

Resolving forces: T cosθ = mg, T sinθ = mω²r. The radius r = L sinθ, where L is the string length. Combining gives ω = √(g / (L cosθ)), and the period T = 2π √(L cosθ / g).

分解力：T cosθ = mg，T sinθ = mω²r。半径 r = L sinθ，其中 L 为绳长。联立可得 ω = √(g / (L cosθ))，周期 T = 2π √(L cosθ / g)。

For a banked curve without friction, the horizontal component of the normal reaction supplies the centripetal force. The optimum banking angle θ satisfies tanθ = v²/(rg), allowing a car to negotiate the curve without lateral friction.

对于无摩擦的倾斜弯道，支持力的水平分量提供向心力。最佳倾斜角 θ 满足 tanθ = v²/(rg)，这使得汽车无需侧向摩擦力即可转弯。

10. Experimental Verification of Centripetal Force | 向心力的实验验证

A classic laboratory setup involves a mass rotated in a horizontal circle using a known hanging weight to provide the centripetal force via a string passing through a tube. By measuring the period and radius, you can verify F = mω²r.

经典实验装置包括：让一个质量在水平面内旋转，用已知的悬挂重物通过穿过管子的绳子提供向心力。通过测量周期和半径，可以验证 F = mω²r。

In this experiment, the tension (equal to the weight of the hanging mass) provides the centripetal force. The rotating mass, its radius, and the time for a fixed number of revolutions are recorded. Angular velocity ω = 2π/T, and the predicted force is mω²r. Agreement within experimental uncertainty confirms the relationship.

在此实验中，张力（等于悬挂重物的重力）提供向心力。记录旋转质量、其半径以及转动固定圈数的时间。角速度 ω = 2π/T，预期向心力为 mω²r。如果在实验不确定度范围内相符，则证实该关系。

Common sources of error include friction in the tube and the difficulty in keeping the radius constant. In improved versions, a force sensor and photogate are used for greater accuracy.

常见误差来源包括管子内的摩擦以及保持半径恒定的困难。在改进的版本中，使用力传感器和光电门来获得更高的精度。

11. Common Mistakes and Exam Tips | 常见错误与应试技巧

Misapplying the equations is a frequent pitfall. Always check whether you are using radius or diameter; ensure v = ωr is applied with r in the same unit system.

误用公式是常见的陷阱。务必检查使用的是半径还是直径；确保应用 v = ωr 时 r 采用一致的单位制。

Many students confuse angular velocity ω (rad s⁻¹) with linear velocity v (m s⁻¹). Remember they relate via v = ωr, but they are different physical quantities.

许多学生混淆角速度 ω (rad s⁻¹) 与线速度 v (m s⁻¹)。记住二者通过 v = ωr 联系，但它们是不同的物理量。

Do not treat centripetal force as an additional force in your free-body diagram. Identify the real forces (tension, weight, normal reaction, friction) and equate their net radial component to mv²/r.

不要在受力分析图中将向心力作为一个额外的力。识别真实力（张力、重力、支持力、摩擦力），并将其径向分量的合力设为等于 mv²/r。

In vertical circle problems, clearly define the positive direction (usually towards the centre). Write separate equations for the top and bottom if needed, and use energy conservation to link speeds.

在竖直圆周问题中，明确正方向（通常指向圆心）。如有必要，分别为最高点和最低点列出方程，并利用能量守恒将速度联系起来。

When a question involves “just completing the circle”, immediately think of the critical condition: tension or normal reaction equals zero at the highest point.

当题目涉及“刚好完成圆周运动”时，立即想到临界条件：在最高点张力或支持力为零。

Show all working clearly, substitute values with units, and express final answers to the appropriate number of significant figures. Practice drawing clear free-body diagrams; they often earn additional marks.

清晰写出所有步骤，代入带单位的数值，并以适当的有效数字位数表示最终答案。练习绘制清晰的受力分析图，它们常能赢得额外分数。

12. Summary of Key Equations | 核心公式总结

The essential equations for circular motion are summarised below. Familiarity with these will allow you to tackle almost any AQA examination problem.

圆周运动的核心公式总结如下。熟悉这些公式将使你能够应对几乎任何 AQA 考试问题。

Quantity 物理量	Equation 公式	Notes 备注
Angular displacement to arc length 角位移与弧长	θ = s / r	θ in radians θ 以弧度计
Angular velocity 角速度	ω = Δθ/Δt ; ω = 2π/T ; ω = 2πf	Unit rad s⁻¹
Linear and angular velocity 线速度与角速度	v = ω r	v perpendicular to radius
Centripetal acceleration 向心加速度	a = v² / r ; a = ω² r	Directed to centre 指向圆心
Centripetal force 向心力	F = m v² / r ; F = m ω² r	Net radial inward force 径向向内的合力
Critical speed at top of vertical circle 竖直圆最高点临界速率	v_min = √(g r)	Tension/ reaction = 0 张力/支持力为零
Conical pendulum period 圆锥摆周期	T = 2π √(L cosθ / g)	θ is angle to vertical θ为与竖直方向夹角

Regular revision of these equations, together with plenty of practice on past paper questions, will build confidence and speed in the exam. Good luck!

定期复习这些公式，并大量练习历年真题，将帮助你在考试中建立信心并提高解题速度。祝你好运！

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Circular Motion | 圆周运动