摘要：刚体定轴转动是经典力学中的重要研究对象，其动能定理揭示了外力矩做功与刚体转动动能变化的本质联系。掌握该定理对理解机械能守恒、分析旋转系统（如飞轮、陀螺仪）的运动特性具有重要意义。

分享乐趣，传播快乐，

增长见识，留下美好。

亲爱的您，

这里是LearingYard学苑！

今天小编为大家带来“PR中的亮度转场”

欢迎您的访问！

Share the fun, spread the joy,

Gain knowledge and leave a good future.

Dear You,

This is LearingYard!

Today, the editor brings you "Brightness transition in PR"

Welcome to visit!

思维导图

Mind mapping

引言

Introduction

刚体定轴转动是经典力学中的重要研究对象，其动能定理揭示了外力矩做功与刚体转动动能变化的本质联系。掌握该定理对理解机械能守恒、分析旋转系统（如飞轮、陀螺仪）的运动特性具有重要意义。

The rotation of a rigid body about a fixed axis is an important subject in classical mechanics. Its kinetic energy theorem reveals the fundamental connection between the work done by external torques and the change in the rotational kinetic energy of the rigid body. Mastering this theorem is of great significance for understanding the conservation of mechanical energy and analyzing the motion characteristics of rotational systems, such as flywheels and gyroscopes.

基础概念回顾

Review of Basic Concepts

1. 刚体定轴转动

1. Rotation of a Rigid Body about a Fixed Axis

刚体内各质点绕同一固定直线（转轴）作圆周运动，角速度矢量方向沿转轴。

Particles within a rigid body perform circular motion around the same fixed line (the axis of rotation), with the direction of the angular velocity vector along the axis.

2. 转动惯量

2. Moment of Inertia

定义式：

Definition:

物理意义：反映刚体对转动的惯性。

Physical Meaning: Reflects the inertia of a rigid body with respect to rotation.

3. 力矩做功

3. Work Done by Torque

Differential form of work done by torque on a rigid body:

动能定理的表述

Statement of the Kinetic Energy Theorem:

定理内容：

Content of the Theorem:

刚体绕固定轴转动时，合外力矩对刚体所做的功等于刚体转动动能的增量。

When a rigid body rotates about a fixed axis, the work done by the net external torque on the body is equal to the increase in the body's rotational kinetic energy.

定理的两种推导方法

Two Methods for Deriving the Theorem

方法1：质点系动能定理推广

Method 1: Generalization of the Kinetic Energy Theorem for a System of Particles

1. 刚体视为质点系，第i个质点的动能：

1. Rigid Body as a System of Particles: Kinetic Energy of the \(i\)-th Particle:

2. 总动能求和：

2. Total Kinetic Energy Summation:

3. 结合力矩做功公式完成推导。

3. Complete the derivation by combining the formula for work done by torque.

方法2：微分形式直接积分

Method 2: Direct Integration of the Differential Form

物理意义与拓展

Physical Significance and Extension

1. 能量视角：力矩做功是能量转化的量度，转动动能是刚体整体运动的能量表现。

1. From an energy perspective: The work done by a torque is a measure of energy transformation, and rotational kinetic energy is the manifestation of the overall motion of a rigid body in terms of energy.

2. 对比平动动能定理

2. Comparison with the Translational Kinetic Energy Theorem

3. 适用条件：仅适用于定轴转动，转轴不可移动。

3. Applicable Conditions: Only applicable to fixed-axis rotation, where the rotational axis must remain stationary.

典型应用实例

Typical Application Examples

案例1：滑轮系统

Case 1: Pulley System

问题：质量为M、半径为R的匀质滑轮，通过轻绳悬挂质量m的物体自由下落，求下落h时的角速度。

Problem: A uniform pulley with mass M and radius R is connected to a hanging mass m via a light rope. Determine the angular velocity of the pulley when the mass m has fallen a distance h.

解答：

answer：

案例2：飞轮制动

Case 2: Flywheel Braking

计算刹车片施加恒定摩擦力矩时，飞轮从角速度ω0到静止所需转过的圈数。

Calculate the number of revolutions the flywheel needs to turn when braking pads apply a constant friction torque, bringing it to rest from an initial angular velocity ω₀.

常见误区提醒

Common Pitfalls Alert

1. 转动动能与质心平动动能的区别（非定轴转动需叠加两者）。

2. 内力矩总功为零，不影响动能变化（类似质点系内力性质）。

1. Rotational kinetic energy differs from the translational kinetic energy of the center of mass. For non - fixed - axis rotation, you must combine both.

2. Internal torques do no net work and thus do not affect kinetic energy change, similar to internal forces in a particle system.

总结

Summary

刚体定轴转动动能定理将力矩的空间累积效应与转动状态变化相联系，是分析旋转系统动力学问题的核心工具之一，后续可结合角动量定理构建完整理论体系。

The kinetic energy theorem of rigid body's fixed - axis rotation links torque's spatial accumulation with rotational state changes. It's a core tool for analyzing rotational system dynamics and can be combined with the angular momentum theorem to form a full theoretical system.

今天的分享就到这里了。

如果你对今天的文章有独特的想法，

欢迎给我们留言，

让我们相约明天，

祝您今天过得开心快乐！

That's all for today's sharing.

If you have a unique idea for today's article,

Welcome to leave us a message,

Let's meet tomorrow,

Have a great day!

本文由LearingYard新学苑，如有侵权，请联系我们。

来源：LearningYard学苑