转动运动学与动力学 — AP 物理 C：力学

1. 什么是转动运动学与动力学？ ★★☆☆☆ ⏱ 3 min

转动运动学与动力学研究刚体的定轴转动，描述转动如何发生、什么引发转动速度的变化。本主题是第5单元转动的核心内容，占AP物理C：力学考试总分的14-20%，同时出现在选择题和自由作答部分，通常会和平动结合组成多模块问题。

按照惯例，逆时针转动定义为正，所有计算中的角量都以弧度为单位。转动运动学描述转动量之间的关系，不涉及转动的起因，而转动动力学将这些量和力矩（力的转动类比）联系起来。本主题直接建立平动和转动力学的类比，简化刚体运动的学习。

2. 转动运动学 ★★☆☆☆ ⏱ 4 min

转动运动学不涉及起因，仅描述转动运动，和平动的直线运动学直接类比。核心物理量包括：

• 角位移$\Delta \theta$：刚体绕转轴转过的角度变化，单位为弧度
• 角速度$\omega = \frac{d\theta}{dt}$：角位移的变化率，单位 rad/s
• 角加速度$\alpha = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2}$：角速度的变化率，单位 rad/s²

对于匀角加速度，我们可以直接推导出和匀加速直线运动类比的运动学公式，只需将$x \to \theta$, $v \to \omega$, $a \to \alpha$替换：

\begin{align*}\omega &= \omega_0 + \alpha t \\\theta &= \omega_0 t + \frac{1}{2}\alpha t^2 \\\omega^2 &= \omega_0^2 + 2\alpha \Delta\theta\end{align*}

对于转动刚体上距离转轴$r$处的任意一点，我们可以将角量和线切向量联系起来，所有点都有指向转轴的向心（径向）加速度：

• Arc length: $s = r\theta$
• Tangential speed: $v = r\omega$
• Tangential acceleration: $a_t = r\alpha$
• Centripetal acceleration: $a_c = r\omega^2$

3. 力矩与转动惯量 ★★★☆☆ ⏱ 3 min

力矩是力的转动类比：就像力引发线运动的变化，力矩是引发转动运动变化的物理量。对于作用点距离转轴$r$的力$F$，力矩的大小为：

\tau = rF\sin\theta = rF_\perp = r_\perp F

Where $\theta$ is the angle between the position vector $\vec{r}$ (from the axis to the point of application) and $\vec{F}$. $F_\perp$ is the component of force perpendicular to $r$, and $r_\perp$ is the perpendicular lever arm from the axis to the line of action of the force. By convention, counterclockwise torque is positive, and clockwise torque is negative.

Rotational inertia (or moment of inertia) is the rotational analog of mass, describing how much torque is needed to produce a given angular acceleration. For a system of discrete masses, $I = \sum m_i r_i^2$, where $r_i$ is the distance of mass $m_i$ from the axis. For a continuous rigid body, $I = \int r^2 dm$. The parallel axis theorem lets you calculate $I$ for any axis parallel to the axis through the center of mass:

I = I_{cm} + Md^2

Where $M$ is the total mass of the object, and $d$ is the distance between the two parallel axes.

4. 转动的牛顿第二定律 ★★★☆☆ ⏱ 4 min

Newton's second law for rotation connects net torque to angular acceleration, analogous to Newton's second law for translation $\sum \vec{F} = m\vec{a}$. For rotation about a fixed axis, the law states:

\sum \tau_{\text{axis}} = I_{\text{axis}} \alpha

This means the net torque on a rigid body about the rotation axis equals the product of the rotational inertia about that axis and the angular acceleration. For problems involving both translation and rotation (e.g., pulley systems with massive pulleys, rolling motion), this law is used alongside Newton's second law for translation, with the relation $a = r\alpha$ connecting linear acceleration of a point on the rigid body to angular acceleration when there is no slipping.