Describing Circular Motion | 描述圆周运动

📚 Describing Circular Motion | 描述圆周运动

Circular motion occurs whenever an object travels along a circular path. In everyday life, a car taking a bend, a satellite orbiting Earth and a stone whirled on a string all involve circular motion. In A-Level physics, we describe this motion using angular quantities such as angular displacement, angular velocity and centripetal acceleration.

当物体沿圆形路径运动时，就发生了圆周运动。在日常生活中，汽车转弯、卫星绕地球运行以及用绳子甩动石子都涉及圆周运动。在 A-Level 物理中，我们使用角位移、角速度和向心加速度等角量来描述这种运动。

1. What is Circular Motion? | 什么是圆周运动？

Circular motion is the motion of an object along the circumference of a circle. If the radius of the path is constant and the speed is constant, the motion is described as uniform circular motion.

圆周运动是物体沿圆周轨迹的运动。如果路径半径不变且速率恒定，这种运动称为匀速圆周运动。

Although the speed is constant in uniform circular motion, the velocity is not constant because the direction of the velocity vector changes at every instant.

在匀速圆周运动中，尽管速率恒定，速度并不是恒定的，因为速度矢量的方向在每一瞬间都在改变。

This continuous change in direction means the object has an acceleration, which is directed towards the centre of the circle.

方向的持续变化意味着物体具有加速度，且该加速度指向圆心。

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2. Radian Measure and Angular Displacement | 弧度制与角位移

Angles in circular motion are usually measured in radians. One radian is defined as the angle subtended at the centre of a circle by an arc whose length equals the radius.

圆周运动中的角度通常用弧度来度量。1 弧度定义为圆弧长度等于半径时所对应的圆心角。

If an object moves along an arc length s on a circle of radius r, the angular displacement θ is given by θ = s / r. This relationship is valid only when θ is expressed in radians.

如果物体在半径为 r 的圆周上经过弧长 s，则角位移 θ 由 θ = s / r 给出。该关系仅在 θ 以弧度表示时成立。

A full circle has arc length 2πr, so an angle of 360° corresponds to 2π radians.

整个圆周的弧长为 2πr，因此 360° 对应 2π 弧度。

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3. Angular Velocity | 角速度

Angular velocity ω measures how quickly an object rotates. The average angular velocity is defined as ω = Δθ / Δt, where Δθ is the angular displacement in a time interval Δt.

角速度 ω 衡量物体转动的快慢。平均角速度定义为 ω = Δθ / Δt，其中 Δθ 是时间间隔 Δt 内的角位移。

The SI unit of angular velocity is the radian per second (rad s⁻¹). For uniform circular motion, ω is constant, and the instantaneous angular velocity equals the average angular velocity.

角速度的国际单位是弧度每秒（rad s⁻¹）。在匀速圆周运动中，ω 保持恒定，瞬时角速度等于平均角速度。

Angular velocity can also be expressed as a vector quantity; its direction is perpendicular to the plane of rotation according to the right-hand rule.

角速度也可以表示为矢量；根据右手定则，其方向垂直于旋转平面。

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4. Relating Linear and Angular Velocity | 线速度与角速度的关系

The linear speed v along the tangent is related to the angular velocity by v = ωr. This follows from the definition θ = s / r because dividing s = rθ by time t gives s/t = r θ/t.

沿切线方向的线速度 v 与角速度的关系为 v = ωr。这可以从 θ = s / r 推出，因为将 s = rθ 除以时间 t 得到 s/t = r θ/t。

In this equation, ω must be measured in radians per second and r is the radius of the circular path. If ω doubles for a fixed radius, the linear speed also doubles.

在该公式中，ω 必须以弧度每秒为单位，r 是圆周路径的半径。如果半径不变，ω 加倍，线速度也会加倍。

The velocity vector v is always tangent to the circle, so its magnitude is v = ωr, but its direction changes continuously.

速度矢量 v 始终沿圆周的切线方向，因此其大小为 v = ωr，但方向不断变化。

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5. Period and Frequency | 周期与频率

The period T is the time taken for one complete revolution. Since one revolution covers an angle of 2π radians, the angular velocity is ω = 2π / T.

周期 T 是物体完成一整圈所需的时间。由于一圈对应 2π 弧度，角速度为 ω = 2π / T。

The frequency f is the number of revolutions per second, so f = 1 / T. Combining these gives the useful relation ω = 2πf.

频率 f 是每秒转动的圈数，因此 f = 1 / T。结合以上关系可得 ω = 2πf。

The linear speed can also be written as v = 2πr / T, which is simply the circumference divided by the period.

线速度也可以表示为 v = 2πr / T，即圆周长除以周期。

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6. Acceleration in Uniform Circular Motion | 匀速圆周运动中的加速度

In uniform circular motion, the speed is constant, yet the velocity vector changes because its direction changes. This change in velocity gives rise to an acceleration towards the centre.

在匀速圆周运动中，速率恒定，但由于速度矢量方向变化，速度矢量仍在改变。这种速度变化产生了指向圆心的加速度。

The acceleration does not change the speed; it only changes the direction of motion. For this reason it is called centripetal acceleration, from Latin meaning ‘seeking the centre’.

该加速度不改变速率，只改变运动方向。因此它被称为向心加速度，源自拉丁语，意为“指向中心”。

If a particle in uniform circular motion has centripetal acceleration aₙ and speed v, these quantities are related by aₙ = v² / r, where r is the radius.

如果做匀速圆周运动的粒子具有向心加速度 aₙ 和速率 v，则它们满足 aₙ = v² / r，其中 r 为半径。

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7. Centripetal Acceleration | 向心加速度

Using the relation v = ωr, the centripetal acceleration can also be expressed as aₙ = ω²r. Both forms aₙ = v² / r and aₙ = ω²r are equivalent.

利用 v = ωr，向心加速度也可表示为 aₙ = ω²r。两种形式 aₙ = v² / r 与 aₙ = ω²r 是等价的。

The direction of aₙ is always towards the centre of the circle, perpendicular to the velocity. The unit of centripetal acceleration is the metre per second squared (m s⁻²).

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