摘要：转动惯量（Moment of Inertia）是描述刚体绕轴转动时惯性大小的物理量，用符号 I 表示。在平动中，质量m衡量物体的惯性；而在转动中，转动惯量扮演了类似的角色——它决定了物体改变自身转动状态的难易程度。

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思维导图

Mind mapping

转动惯量的基本概念

The Basic Concept of Moment of Inertia

转动惯量（Moment of Inertia）是描述刚体绕轴转动时惯性大小的物理量，用符号 I 表示。在平动中，质量m衡量物体的惯性；而在转动中，转动惯量扮演了类似的角色——它决定了物体改变自身转动状态的难易程度。

Moment of inertia is a physical quantity that describes the inertia of a rigid body rotating about an axis,denoted by I.While mass m measures inertia in translation,moment of inertia plays a similar role in rotation,determining the difficulty of changing the body’s rotational state.

数学定义：

对于离散质点系，转动惯量为各质点质量与其到转轴距离平方的乘积之和：

For a system of discrete particles,the moment of inertia is the sum of the products of each particle's mass and the square of its distance to the axis of rotation.

对于连续分布的刚体，转动惯量通过积分计算：

For a continuous rigid body,the moment of inertia is calculated through integration.

其中r是质元dm到转轴的垂直距离。

Where r is the perpendicular distance from the mass element dm to the axis of rotation.

转动惯量的物理意义

The physical significance of moment of inertia

1.转动惯量与角加速度的关系

The relationship between moment of inertia and angular acceleration.

由转动定律：

可知，转动惯量越大，相同力矩下角加速度越小，物体越难改变转动状态。

According to the law of rotation,the larger the moment of inertia,the smaller the angular acceleration under the same torque,and the more difficult it is to change the rotational state of the object.

2.质量分布的影响

The Impact of Mass Distribution

转动惯量不仅与总质量有关，还与质量相对转轴的分布密切相关。例如：质量离转轴越远，转动惯量越大；相同质量的圆环与圆盘相比，圆环的转动惯量更大。

Moment of inertia depends not only on the total mass but also on the distribution of mass relative to the axis of rotation.For example,the farther the mass is from the axis,the greater the moment of inertia.Compared to a disk of the same mass,a ring has a larger moment of inertia.

常见刚体的转动惯量公式

Formulas for the moment of inertia of common rigid bodies

1.细杆（转轴通过中心）

Thin rod(axis through center)

2. 圆盘/圆柱（转轴沿几何轴）

Disk/Cylinder(axis along the geometric axis)

3. 实心球（转轴通过球心）

Solid sphere(axis through the center)

4. 薄壁空心球（转轴通过球心）

Thin-walled hollow sphere(axis through the center)

平行轴定理与垂直轴定理

Parallel-axis theorem and perpendicular-axis theorem

1. 平行轴定理

Parallel-axis theorem

若刚体对某一转轴的转动惯量为I，则对与之平行的另一轴（间距为d）的转动惯量为：

If the moment of inertia of a rigid body about one axis is I,then the moment of inertia about a parallel axis(separated by a distance d is:

应用：快速计算非对称转轴的转动惯量。

Application:Quickly calculate the moment of inertia for non-symmetric axes.

2.垂直轴定理（仅适用于薄板）

Perpendicular-axis theorem(applicable only to thin plates)

薄板对平面内两垂直轴的转动惯量之和等于对垂直于平面且过交点的轴的转动惯量：

For a thin plate,the sum of the moments of inertia about two perpendicular axes in the plane equals the moment of inertia about the axis perpendicular to the plane and passing through their intersection.

转动惯量的实际应用

Practical applications of moment of inertia

1. 机械工程

Mechanical Engineering

飞轮设计：增大转动惯量可储存更多动能，稳定机械运动；

车辆轮毂：优化质量分布以减少能量损耗。

Flywheel design:Increasing moment of inertia stores more kinetic energy and stabilizes mechanical motion.

Vehicle wheels:Optimizing mass distribution reduces energy loss.

2. 体育运动

Sports

体操运动员通过收紧身体（减小I）提高旋转速度；

跳水运动员伸展身体（增大I）以延缓翻转。

Gymnasts increase their rotation speed by tucking their bodies(reducing I);divers extend their bodies(increasing I)to slow down the flip.

3. 天体物理学

Astrophysics

行星自转惯量影响其形状和内部结构分析；

中子星的转动惯量可用于研究其物质状态方程。

The rotational inertia of planets affects their shape and internal structure analysis;the rotational inertia of neutron stars can be used to study their equation of state.

总结

Summary

转动惯量是分析刚体转动问题的核心物理量，其大小由质量分布和转轴位置共同决定。掌握常见几何体的转动惯量公式及定理（如平行轴定理），能有效简化复杂系统的计算。在工程和自然科学中，转动惯量的优化与控制对提高系统性能具有重要意义。

Moment of inertia is a key physical quantity for analyzing rigid body rotation,determined by mass distribution and axis position.Mastering formulas and theorems(e.g.,parallel-axis theorem)for common geometries can simplify complex system calculations.In engineering and natural sciences,optimizing and controlling moment of inertia is crucial for enhancing system performance.

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部分参考内容来自百度

翻译来源：Kimi翻译

来源：LearningYard学苑