Motion in Polar Coordinates | 极坐标下的运动描述

📚 Motion in Polar Coordinates | 极坐标下的运动描述

Describing motion in polar coordinates is a powerful technique in IB Mathematics, especially when dealing with circular or spiral paths. Instead of using fixed x and y axes, we locate a point by its distance from the origin and the angle it makes with a reference direction.

在极坐标中描述运动是IB数学中非常实用的技巧，特别适合处理圆周或螺旋轨迹。我们不再使用固定的x轴和y轴，而是通过点到原点的距离以及与参考方向之间的夹角来确定位置。

1. Polar Coordinate Basics | 极坐标基础

In polar coordinates, a point P is represented by (r, θ), where r is the radial distance from the pole (origin) and θ is the polar angle measured from the positive x-axis. The conversion to Cartesian coordinates is:

在极坐标系中，点P用(r, θ)表示，其中r是到极点（原点）的径向距离，θ是从正x轴起逆时针旋转的极角。转换为直角坐标的关系为：

x = r cos θ, y = r sin θ, r² = x² + y², tan θ = y / x

When describing motion, both r and θ can depend on time t. Thus we write r = r(t) and θ = θ(t).

描述运动时，r和θ都可以随时间t变化，因此我们写成r = r(t) 和 θ = θ(t)。

2. Unit Vectors in Polar Coordinates | 极坐标中的单位矢量

The key to analyzing motion in polar coordinates is using the two unit vectors: the radial unit vector e_r and the transverse (angular) unit vector e_θ.

分析极坐标中运动的关键是使用两个单位矢量：径向单位矢量e_r和横向（角向）单位矢量e_θ。

These are defined as:

它们的定义如下：

e_r = cos θ i + sin θ j

e_θ = -sin θ i + cos θ j

Here i and j are the standard Cartesian unit vectors. Notice that e_θ is obtained by rotating e_r by 90 degrees counterclockwise.

这里i和j是标准直角坐标单位矢量。注意e_θ由e_r逆时针旋转90度得到。

3. Derivatives of Unit Vectors | 单位矢量的导数

To find velocity and acceleration, we must differentiate the unit vectors with respect to time. Using the chain rule:

为了求速度和加速度，我们必须对单位矢量关于时间求导。利用链式法则：

d(e_r)/dt = θ̇ e_θ

d(e_θ)/dt = -θ̇ e_r

Here θ̇ = dθ/dt is the angular velocity. These results are fundamental: the radial unit vector changes in the transverse direction, and the transverse unit vector changes in the negative radial direction.

这里θ̇ = dθ/dt是角速度。这些结果非常重要：径向单位矢量的变化方向是横向，而横向单位矢量的变化方向是负径向。

4. Position Vector | 位置矢量

The position vector of a particle in polar coordinates is simply:

质点在极坐标中的位置矢量可以简洁地写成：

r = r e_r

This means the particle is at a distance r from the origin along the direction e_r. There is no transverse component of position because e_θ points perpendicular to the radial direction.

这意味着质点沿e_r方向距离原点为r。位置没有横向分量，因为e_θ垂直于径向方向。

5. Velocity in Polar Coordinates | 极坐标中的速度

Differentiating the position vector with respect to time using the product rule and the derivative of e_r gives:

对位置矢量关于时间求导，使用乘积法则以及e_r的导数，得到：

v = ṙ e_r + r θ̇ e_θ

Thus the velocity has two components:

因此速度具有两个分量：

Radial component: v_r = ṙ, the rate at which the distance from the origin changes.
Transverse component: v_θ = r θ̇, the speed due to rotation around the origin.

径向分量：v_r = ṙ，表示距原点距离的变化快慢。

横向分量：v_θ = r θ̇，表示绕原点旋转产生的速度。

6. Acceleration in Polar Coordinates | 极坐标中的加速度

Differentiating the velocity vector again, and carefully applying the product rule to both terms, yields the acceleration:

再次对速度矢量求导，并仔细对两项应用乘积法则，得到加速度：

a = (r̈ – r θ̇²) e_r + (r θ̈ + 2 ṙ θ̇) e_θ

The radial component contains r̈ and the centripetal term -r θ̇². The transverse component contains r θ̈ and the Coriolis term 2 ṙ θ̇.

径向分量包含r̈和向心项-r θ̇²。横向分量包含r θ̈和科里奥利项2 ṙ θ̇。

In uniform circular motion, r is constant, so ṙ = 0 and r̈ = 0, giving a = -r θ̇² e_r, which points toward the center.

在匀速圆周运动中，r为常数，因此ṙ = 0且r̈ = 0，得到a = -r θ̇² e_r，指向圆心。

7. Uniform Circular Motion | 匀速圆周运动

For a particle moving on a circle of radius R with constant angular speed ω, we have r = R and θ = ωt + θ₀. Then:

对于半径为R、角速度恒为ω的圆周运动，有r = R，θ = ωt + θ₀。于是：

v = R ω e_θ

a = -R ω² e_r

This matches the familiar results: speed = Rω and centripetal acceleration = Rω². The acceleration vector always points towards the center of the circle.

这与我们熟悉的结果一致：速率 = Rω，向心加速度 = Rω²。加速度矢量始终指向圆心。

8. Angular Velocity and Angular Acceleration | 角速度与角加速度

In polar motion problems, θ̇ represents the angular velocity and θ̈ represents the angular acceleration. These are often denoted by ω and α respectively.

在极坐标运动问题中，θ̇表示角速度，θ̈表示角加速度。它们通常分别用ω和α表示。

When θ̈ = 0, the angular velocity is constant, and the transverse acceleration simplifies to 2 ṙ θ̇, which is nonzero only if the radial distance is changing.

当θ̈ = 0时，角速度为常数，横向加速度简化为2 ṙ θ̇，只有在径向距离变化时才不为零。

Example: if r = t² and θ = 3t, then θ̇ = 3 and θ̈ = 0. The radial distance increases steadily, causing a transverse acceleration even though the angular speed is constant.

例如：若r = t²且θ = 3t，则θ̇ = 3，θ̈ = 0。径向距离持续增大，即使角速度恒定，也会产生横向加速度。

9. Motion along a Spiral | 螺旋运动

A common application is the Archimedean spiral, where r = aθ. Suppose a particle moves along this spiral with constant angular speed ω, so θ = ωt.

一个常见的应用是阿基米德螺旋线，满足r = aθ。假设质点以恒定角速度ω沿此螺旋运动，即θ = ωt。

Then r = aωt, ṙ = aω, and r̈ = 0. The velocity components are:

则r = aωt，ṙ = aω，r̈ = 0。速度分量为：

v = aω e_r + aω² t e_θ

The acceleration components are:

加速度分量为：

a = -aω³ t² e_r + 2aω² e_θ

Notice how the radial acceleration grows with t² due to the increasing curvature as the spiral expands.

注意径向加速度随t²增大，这是因为螺旋扩展时曲率不断变化。

10. Relating Polar and Cartesian Descriptions | 极坐标与直角坐标描述的联系

Sometimes it is necessary to switch between polar and Cartesian descriptions. Given x(t) and y(t), we can find r and θ:

有时需要在极坐标和直角坐标描述之间转换。已知x(t)和y(t)，可求出r和θ：

r = √(x² + y²), θ = arctan(y/x)

Then the polar velocity components can be computed from ṙ = (x ẋ + y ẏ) / r and θ̇ = (x ẏ – y ẋ) / r².

然后可由ṙ = (x ẋ + y ẏ) / r和θ̇ = (x ẏ – y ẋ) / r²计算极坐标速度分量。

Conversely, if v_r and v_θ are known, the Cartesian velocity components are:

反之，如果已知v_r和v_θ，直角坐标速度分量为：

ẋ = v_r cos θ – v_θ sin θ

ẏ = v_r sin θ + v_θ cos θ

11. Worked Example: Elliptical Motion | 例题：椭圆运动

Consider a particle moving such that r = a(2 + cos t) and θ = t. Find the radial acceleration at t = 0.

考虑一个质点，其运动满足r = a(2 + cos t)和θ = t。求t = 0时径向加速度。

First compute derivatives:

首先计算导数：

ṙ = -a sin t, r̈ = -a cos t, θ̇ = 1, θ̈ = 0

At t = 0: r = 3a, ṙ = 0, r̈ = -a, θ̇ = 1. Therefore:

在t = 0时：r = 3a，ṙ = 0，r̈ = -a，θ̇ = 1。因此：

a_r = r̈ – r θ̇² = -a – 3a = -4a

The negative sign indicates the radial acceleration points toward the origin (in the -e_r direction). The transverse acceleration is a_θ = r θ̈ + 2 ṙ θ̇ = 0.

负号表示径向加速度指向原点（沿-e_r方向）。横向加速度为a_θ = r θ̈ + 2 ṙ θ̇ = 0。

12. Exam Tips and Common Mistakes | 考试技巧与常见错误

To succeed in polar motion questions, keep these points in mind:

要在极坐标运动问题中取得好成绩，请谨记以下几点：

Always remember that e_r and e_θ change with time. Never treat them as fixed vectors.
Always记住e_r和e_θ随时间变化。切勿将它们视为固定矢量。
Use the full acceleration formula, including the -r θ̇² and 2 ṙ θ̇ terms. These terms are often missed.
使用完整加速度公式，包括-r θ̇²和2 ṙ θ̇项。这些项经常被遗漏。
Check units: θ̇ must be in radians per unit time, not degrees.
检查单位：θ̇必须以弧度/单位时间表示，而不是度。
For uniform circular motion, do not forget the acceleration is purely radial and inward.
对于匀速圆周运动，不要忘记加速度是纯径向且向内的。
When converting from Cartesian to polar, use atan2 or quadrant checks to get the correct θ.
从直角坐标转换为极坐标时，使用atan2或象限判断来获得正确的θ。

Practicing differentiation of unit vectors and applying the product rule correctly will help you avoid calculation errors.

练习单位矢量的微分并正确应用乘积法则，将有助于避免计算错误。

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Motion in Polar Coordinates | 极坐标下的运动描述