AQA Physics Topic Test: Circular and Periodic Motion | AQA 物理专题测试：圆周运动与周期运动

📚 AQA Physics Topic Test: Circular and Periodic Motion | AQA 物理专题测试：圆周运动与周期运动

This comprehensive topic test covers the essential concepts of circular and periodic motion for the OxfordAQA International A Level Physics specification. It combines focused revision notes, exam-style questions, and fully worked solutions to help you build mastery in uniform circular motion, centripetal acceleration and force, simple harmonic motion, energy transformations, damping, and resonance.

本综合专题测试覆盖牛津AQA国际A Level物理大纲中圆周运动与周期运动的核心概念，包含精要复习笔记、考试风格试题及完整答案解析，帮助你全面掌握匀速圆周运动、向心加速度与向心力、简谐运动、能量转化、阻尼和共振等关键内容。

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1. Angular Displacement and Angular Velocity | 角位移与角速度

Angular displacement, θ, is the angle swept out by a radius vector as an object moves around a circle. It is measured in radians (rad), where one complete revolution equals 2π radians. The relationship between arc length s, radius r, and angular displacement is given by s = rθ, provided θ is expressed in radians. To convert from degrees to radians, multiply by π/180; for example, 360° = 2π rad, 180° = π rad, and 90° = π/2 rad.

角位移 θ 是物体沿圆周运动时半径矢量所扫过的角度，单位为弧度（rad），一整圈对应 2π 弧度。弧长 s、半径 r 与角位移 θ 的关系为 s = rθ，前提是 θ 以弧度表示。度与弧度的换算公式为：数值乘以 π/180，例如 360° = 2π rad，180° = π rad，90° = π/2 rad。

Angular velocity, ω, is defined as the rate of change of angular displacement: ω = Δθ/Δt. Its SI unit is radians per second (rad s⁻¹). For uniform circular motion, the angular velocity is constant, and it connects directly to the period T (the time for one complete revolution) and the frequency f (the number of revolutions per second) through ω = 2π/T = 2πf. Since frequency and period are reciprocals, T = 1/f, this relationship is extremely important in exam calculations.

角速度 ω 定义为角位移的变化率：ω = Δθ/Δt，其国际单位制单位为弧度每秒（rad s⁻¹）。在匀速圆周运动中，角速度为恒定值，且与周期 T（完成一整圈所需时间）和频率 f（每秒转数）直接关联：ω = 2π/T = 2πf。由于频率与周期互为倒数，T = 1/f，这一关系在考试计算中极为重要。

θ = s/r, ω = Δθ/Δt = 2π/T = 2πf

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2. Linear Speed and Angular Speed Relation | 线速度与角速度的关系

For an object moving in a circle of radius r, the instantaneous linear speed v along the tangent to the circle is related to the angular velocity by v = rω. This result follows directly from differentiating the arc length equation s = rθ with respect to time. Because the velocity vector is always tangent to the circular path, it is frequently called the tangential velocity. Its magnitude may remain constant in uniform circular motion, but the direction changes continuously at every instant.

对于半径为 r 的圆周运动物体，沿圆周切线方向的瞬时线速度 v 与角速度 ω 的关系为 v = rω。该结果直接由弧长方程 s = rθ 对时间求导得出。由于速度矢量始终沿圆周的切线方向，故通常称为切线速度。在匀速圆周运动中，速度大小保持不变，但速度方向每时每刻都在发生改变。

A key exam point is that even when an object moves at constant speed around a circle, it is still accelerating because velocity is a vector quantity and its direction is changing. The faster the angular speed or the larger the radius, the greater the linear speed. For example, in a rotating disc, points further from the centre move with higher linear speed than points near the centre, even though every point on the disc shares the same angular velocity.

一个关键考点是：即使物体以恒定速率绕圆周运动，它仍然具有加速度，因为速度是矢量，方向在不断变化。角速度越大或半径越大，线速度就越大。例如，在旋转圆盘上，离圆心越远的点线速度越大，而圆盘上所有点的角速度相同。

v = rω

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3. Centripetal Acceleration | 向心加速度

Since the velocity direction changes continuously during circular motion, there must be an acceleration. For uniform circular motion, this acceleration is directed radially towards the centre of the circle and is called the centripetal acceleration. Its magnitude is given by a = v²/r, which can also be expressed as a = rω² by substituting v = rω. The direction of centripetal acceleration is always perpendicular to the velocity vector, meaning it changes the direction of velocity without changing its magnitude.

由于圆周运动中速度方向连续改变，因此必然存在加速度。在匀速圆周运动中，该加速度沿半径指向圆心，称为向心加速度。其大小由 a = v²/r 给出，代入 v = rω 后也可写为 a = rω²。向心加速度的方向始终垂直于速度矢量，它只改变速度方向而不改变速度大小。

It is helpful to visualise the velocity vector rotating as the object moves. Over a small time interval, the change in velocity Δv points towards the centre of the circle; dividing by the time interval gives the instantaneous acceleration. The centripetal acceleration increases with speed squared but decreases as the radius increases. A tight bend at high speed therefore produces a very large acceleration, which is why racing tracks have wide corners.

可以将速度矢量想象为随物体运动而旋转。在极小的时间间隔内，速度变化量 Δv 指向圆心；除以时间间隔即得到瞬时加速度。向心加速度随速度的平方增大而增大，随半径增大而减小。因此，高速急转弯会产生非常大的加速度，这也解释了为什么赛车赛道需要宽大的弯道。

a = v²/r = rω²

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4. Centripetal Force | 向心力

Newton’s second law states that a net force is required to produce any acceleration. For circular motion, the resultant force that causes the centripetal acceleration is called the centripetal force, F = ma = mv²/r = mrω². It acts radially inwards towards the centre of the circle. Centripetal force is not a new or separate type of force; rather, it is the name given to the resultant force that must be directed towards the centre to sustain circular motion.

牛顿第二定律指出，产生任何加速度都需要合外力。对于圆周运动，产生向心加速度的合力称为向心力，F = ma = mv²/r = mrω²，方向沿半径指向圆心。向心力并非一种新型或独立的力，而是维持圆周运动所必需的、指向圆心的合力。

In different physical situations, the centripetal force is provided by different real forces. For a car rounding a flat bend, friction between the tyres and the road supplies the force. For a satellite in orbit, gravity is the centripetal force. For a ball whirled on a string, tension in the string provides it. For an aircraft performing a loop, the lift force contributes. Common exam contexts also include a mass on a turntable, a cyclist leaning into a corner, and the banking of roads to reduce reliance on friction.

在不同的物理情境中，向心力由不同的实际力提供：汽车在水平弯道转弯时，轮胎与路面之间的摩擦力提供向心力；轨道上的卫星由万有引力提供向心力；绳子拴球旋转时，绳中张力提供向心力；飞机做筋斗运动时，升力提供向心力。常见的考试情境还包括转盘上的物块、骑车人过弯时倾斜车身，以及路面外侧超高以减小对摩擦力的依赖。

F = mv²/r = mrω²

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5. Simple Harmonic Motion Definition | 简谐运动定义

Simple Harmonic Motion (SHM) is a type of periodic oscillation in which the acceleration of the object is directly proportional to its displacement from an equilibrium position and is always directed towards that equilibrium position. This defining condition is written mathematically as a = -ω²x, where x is the displacement from equilibrium, ω is the angular frequency, and the negative sign shows that acceleration and displacement are in opposite directions.

简谐运动（SHM）是一种周期性振动，其特点是物体的加速度与偏离平衡位置的位移成正比，且始终指向平衡位置。这一核心条件用数学式表示为 a = -ω²x，其中 x 为相对于平衡位置的位移，ω 为角频率，负号表示加速度与位移方向相反。

When an object is pulled away from equilibrium, the restoring force pushes or pulls it back, but the object overshoots equilibrium due to inertia, causing it to oscillate. Examples include a mass attached to a spring, a simple pendulum swinging through small angles, and the vibration of a tuning fork. Two conditions must be satisfied for SHM: the acceleration must be proportional to displacement, and it must oppose displacement. If either condition fails, the motion is not simple harmonic.

当物体偏离平衡位置时，恢复力使其回到平衡位置，但由于惯性，物体会越过平衡位置，从而产生振动。常见例子包括连接弹簧的物块、小角度摆动的单摆，以及音叉的振动。发生简谐运动必须满足两个条件：加速度与位移成正比，且加速度方向与位移方向相反。如果任一条件不满足，则该运动不是简谐运动。

a = -ω²x

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6. SHM Equations and Graphical Analysis | 简谐运动方程与图像分析

If an object begins at its maximum displacement A (the amplitude) at time t = 0, its displacement at any later time is given by x = A cos(ωt), where ω = 2πf = 2π/T. Alternatively, if it starts at equilibrium, the correct expression is x = A sin(ωt). Differentiating the displacement equation with respect to time gives the velocity v = -Aω sin(ωt), and differentiating again gives the acceleration a = -Aω² cos(ωt) = -ω²x.

如果物体在 t = 0 时从最大位移 A（即振幅）处开始运动，则在任意时刻 t 的位移为 x = A cos(ωt)，其中 ω = 2πf = 2π/T。如果物体从平衡位置开始运动，则应使用 x = A sin(ωt)。将位移方程对时间求导可得速度 v = -Aω sin(ωt)，再次求导可得加速度 a = -Aω² cos(ωt) = -ω²x。

The maximum speed is Aω and occurs as the object passes through the equilibrium position, where displacement is zero. The maximum acceleration

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