📚 Circular and Periodic Motion | 圆周运动与周期运动
In AQA International A-level Physics, circular and periodic motion brings together rotational kinematics, centripetal force, and the mathematics of simple harmonic motion (SHM). Understanding these ideas is essential for analysing objects moving in circles and oscillating systems such as pendulums and mass-spring systems.
在 AQA 国际 A-level 物理中，圆周运动与周期运动将转动运动学、向心力以及简谐运动（SHM）的数学描述结合在一起。理解这些概念对于分析圆周运动物体以及摆和弹簧振子等振动系统至关重要。
1. Describing Circular Motion | 描述圆周运动
An object moving in a circle at constant speed is said to be in uniform circular motion. Even though the speed is constant, the velocity is changing because the direction of motion changes continuously. The object is therefore accelerating towards the centre of the circle.
物体以恒定速率沿圆周运动时，称为匀速圆周运动。尽管速率保持不变，但由于运动方向不断改变，速度矢量在变化。因此，物体具有指向圆心的加速度。
The angle turned through per unit time is called the angular displacement, measured in radians. One radian is the angle subtended at the centre of a circle by an arc length equal to the radius.
单位时间内转过的角度称为角位移，单位为弧度。一弧度是指圆弧长度等于半径时所对应的圆心角。
θ = s / r
For one complete revolution, the arc length s is the circumference 2πr, so the angular displacement is 2π radians.
对于完整的一圈，弧长 s 等于周长 2πr，因此角位移为 2π 弧度。
2. Angular Velocity and Period | 角速度与周期
Δt. Since one complete revolution is 2π radians, ω is related to the period T by ω = 2π/T.
角速度 ω 定义为角位移的变化率。对于匀速圆周运动，ω = Δθ/Δt。由于完整一圈为 2π 弧度，角速度与周期 T 的关系为 ω = 2π/T。
ω = 2π / T = 2πf
radius of the circle. The frequency f is the number of revolutions per second, so f = 1/T.
圆周上一点的线速度 v 与角速度的关系为 v = ωr，其中 r 为圆周半径。频率 f 是每秒转过的圈数，因此 f = 1/T。
v = ωr
Angular velocity is measured in radians per second (rad s⁻¹), while linear speed is measured in metres per second (m s⁻¹).
角速度的单位是弧度每秒（rad s⁻¹），而线速度的单位是米每秒（m s⁻¹）。
3. Centripetal Acceleration | 向心加速度
eration is called centripetal acceleration. Its magnitude is given by a = v²/r or a = ω²r.
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。其大小由 a = v²/r 或 a = ω²r 给出。
a = v² / r = ω²r
Although the speed is constant, the acceleration is not zero because the direction of velocity changes. The acceleration vector always points towards the centre, perpendicular to the velocity vector.
尽管速率恒定，加速度并不为零，因为速度方向在变化。加速度矢量始终指向圆心，并与速度矢量垂直。
tal acceleration are equivalent. For a given radius, doubling the angular velocity increases the centripetal acceleration by a factor of four.
利用 v = ωr，向心加速度的两种表达式是等价的。对于给定的半径，角速度加倍会使向心加速度增大到原来的四倍。
4. Centripetal Force | 向心力
According to Newton' celeration. This force is called centripetal force, and its magnitude is F = mv²/r = mω²r.
根据牛顿第二定律，必须有指向圆心的合力来产生向心加速度。这个力称为向心力，其大小为 F = mv²/r = mω²r。
F = mv² / r = mω²r
Centripetal force is not a new type of force. It can be provided by tension, friction, gravitational attraction, or the normal contact force, depending on the situation.
向心力不是一种新型力。它可以由拉力、摩擦力、万有引力或支持力等提供，视具体情境而定。
In many problems, you should first identify the force or forces acting towards the centre of the circle, then equate their resultant to mv²/r or mω²r.
在许多问题中，你应首先确定指向圆心的一个或多个力，然后将它们的合力与 mv²/r 或 mω²r 相等。
5. Examples of Circular Motion | 圆周运动实例
A car turning on a flat road relies on friction between the tyres and the road to provide the centripetal force. If the required centripetal force exceeds the maximum friction, the car skids.
汽车在水平路面上转弯时，依靠轮胎与路面之间的摩擦力提供向心力。如果所需的向心力超过最大静摩擦力，汽车就会打滑。
For a ball whirled on a string in a horizontal circle, the tension in the string supplies the centripetal force. If the string breaks, the ball moves off along the tangent with constant velocity, because no resultant force acts on it horizontally.
对于用绳子在水平面内旋转的小球，绳子的拉力提供向心力。如果绳子断裂，小球将沿切线方向以恒定速度飞出，因为在水平方向上不再有合力作用于它。
A satellite orbiting the Earth experiences gravitational attraction towards the planet's centre, which acts as the centripetal force required for circular motion. The same principle applies to planets orbiting the Sun.
绕地球运行的卫星受到指向地心的万有引力作用，这个引力充当圆周运动所需的向心力。同样的原理也适用于绕太阳运行的行星。
6. Introduction to Periodic Motion and SHM | 周期运动与简谐运动导论
Periodic motion is any motion that repeats itself after a fixed time interval called the period T. Simple harmonic motion (SHM) is a special type of oscillation in which the restoring force is directly proportional to the displacement from equilibrium and always acts towards the equilibrium position.
周期运动是指每隔固定时间间隔（周期 T）重复的运动。简谐运动（SHM）是一种特殊的振动，其回复力与离开平衡位置的位移成正比，并且始终指向平衡位置。
F = -kx
x, where x is the displacement and ω is the angular frequency. The negative sign shows that acceleration and displacement are always in opposite directions.
简谐运动的定义方程为 a = -ω²x，其中 x 为位移，ω 为角频率。负号表明加速度与位移始终方向相反。
a = -ω²x
For a mass-spring system, the angular frequency is ω = √k/m, where k is the spring constant and m is the mass.
对于弹簧振子，角频率为 ω = √k/m，其中 k 为弹簧劲度系数，m 为质量。
7. Kinematics of SHM | 简谐运动运动学
g on the starting position. A is the amplitude, the maximum displacement from equilibrium.
做简谐运动的物体的位移可以用 x = A cos(ωt) 或 x = A sin(ωt) 描述，具体形式取决于初始位置。A 为振幅，即离开平衡位置的最大位移。
x = A cos(ωt)
The velocity in SHM varies with displacement according to v = ±ω√A² - x² quilibrium position where x = 0, and the speed is zero at the turning points where x = ±A.
简谐运动中的速度随位移变化，关系为 v = ±ω√A² - x²。最大速度出现在平衡位置（x = 0），而在位移最大处（x = ±A）速度为零。
v = ±ω√A² - x²
The period T of SHM is independent of amplitude for an ideal oscillator. This property is called isochronism and is a key feature of simple harmonic motion.
对于理想振子，简谐运动的周期 T 与振幅无关。这一特性称为等时性，是简谐运动的重要特征。
8. Energy in SHM | 简谐运动中的能量
In SHM, energy continuously transfers between kinetic energy and potential energy. Kinetic energy is maximum at the equilibrium position, while potential energy is maximum at the extreme positions.
在简谐运动中，能量在动能和势能之间不断转化。动能最大值出现在平衡位置，而势能最大值出现在最大位移处。
where k is the spring constant. This total energy remains constant if there is no damping.
对于弹簧振子，总能量 E = ½mω²A² = ½kA²，其中 k 为弹簧劲度系数。如果没有阻尼，总能量保持不变。
E = ½mω²A² = ½kA²
At the equilibrium position, all the energy
Published by TutorHao | A-Level Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图