Rotational-Linear Parallels

索引转动惯量概念

HyperPhysics ***** 力学 ***** 转动

R Nave

Conservation of linear momentum dictates that when a mass strikes an equal mass at rest and sticks to it, the combination must move at half the velocity, because the product of mass and velocity must remain constant.

Rotational-Linear Parallels

旋转-线性类比

动量守恒定律指出，当一个质量撞击一个静止的等质量物体并粘附在一起时，组合体必须以原来速度的一半运动，因为质量与速度的乘积必须保持不变。

Using a string through a tube, a mass is moved in a horizontal circle with angular velocity ω. If the string is pulled down so that the radius is half the original radius, then conservation of angular momentum dictates that the ball must have four times the angular velocity. This is because the product of moment of inertia and angular velocity must remain constant, and halving the radius reduces the moment of inertia by a factor of four.

用一根通过管子的绳子，使质量在水平面上以角速度ω做圆周运动。如果绳子被向下拉，使半径变为原来的一半，那么角动量守恒定律表明球必须有四倍的角速度。这是因为转动惯量与角速度的乘积必须保持不变，而半径减半会使转动惯量减少四倍。

With the appropriate balance of force, a circular orbit can be produced by a force acting toward the center. Acting perpendicular to the velocity, it provides the necessary centripetal force to keep it in a circle.

在适当的力的作用下，向心力可以使物体产生圆周运动。该力垂直于速度，提供维持圆周运动所需的向心力。

If a spinning wheel and axle is supported by one end of the axle, then the torque produced by the weight of the wheel and axle produces a torque that is perpendicular to the angular momentum of the wheel. This changes its direction but not its magnitude, causing the tip of the axle to trace out a circle. This is called precession, and is analogous to the orbit of a mass under a central force.

如果一个旋转的轮轴被轴的一端支撑，那么由轮轴重量产生的力矩会产生一个与轮轴角动量垂直的力矩。这会改变其方向但不改变其大小，导致轴的末端轨迹形成一个圆。这种现象称为进动，类似于质量在中心力作用下的轨道运动。

索引转动惯量概念

HyperPhysics ***** 力学 ***** 转动

R Nave

Development for point mass

Moment of Inertia

转动惯量

Moment of inertia is the name given to rotational inertia, the rotational analog of mass for linear motion. It appears in the relationships for the dynamics of rotational motion. The moment of inertia must be specified with respect to a chosen axis of rotation. For a point mass, the moment of inertia is just the mass times the square of perpendicular distance to the rotation axis, I = mr². That point mass relationship becomes the basis for all other moments of inertia since any object can be built up from a collection of point masses.

转动惯量是旋转惯性的名称，它是线性运动中质量的旋转类比。它出现在旋转运动动力学的关系中。转动惯量必须相对于所选的旋转轴来指定。对于质点，转动惯量就是质量乘以该质点到旋转轴垂直距离的平方，I = mr²。这个质点关系成为所有其他转动惯量关系的基础，因为任何物体都可以由一系列质点组成。

Common forms

常见形式

Examples

示例

General form

一般形式

点质量的发展

索引转动惯量概念

HyperPhysics ***** 力学 ***** 转动

R Nave

Common Moments of Inertia

常见惯性 moment

索引转动惯量概念

HyperPhysics ***** 力学 ***** 转动

R Nave

Moments of inertia for common forms

Moment of Inertia Examples

转动惯量示例

Moment of inertia is defined with respect to a specific rotation axis. The moment of inertia of a point mass with respect to an axis is defined as the product of the mass times the distance from the axis squared. The moment of inertia of any extended object is built up from that basic definition. The general form of the moment of inertia involves an integral.

转动惯量是相对于特定的旋转轴定义的。一个质点相对于某轴的转动惯量定义为质量与该质点到轴的距离平方的乘积。任何延伸物体的转动惯量都是基于这个基本定义构建起来的。转动惯量的一般形式涉及一个积分。

常见形状的惯性矩

Where moment of inertia appears in physical quantities

转动惯量出现在物理量中

索引转动惯量概念

HyperPhysics ***** 力学 ***** 转动

R Nave

Moment of Inertia, General Form

转动惯量，一般形式

Since the moment of inertia of an ordinary object involves a continuous distribution of mass at a continually varying distance from any rotation axis, the calculation of moments of inertia generally involves calculus, the discipline of mathematics which can handle such continuous variables. Since the moment of inertia of a point mass is defined by

由于普通物体的转动惯量涉及质量在不断变化的距旋转轴的距离上的连续分布，因此计算转动惯量通常需要微积分，这门数学学科能够处理此类连续变量。由于点质量的转动惯量定义为

then the moment of inertia contribution by an infinitesmal mass element dm has the same form. This kind of mass element is called a differential element of mass and its moment of inertia is given by

然后，一个微小质量元dm对转动惯量的贡献具有相同的形式。这种质量元被称为质量的微分元，其转动惯量由以下公式给出

Note that the differential element of moment of inertia dI must always be defined with respect to a specific rotation axis. The sum over all these mass elements is called an integral over the mass.

需要注意的是，转动惯量的微元dI必须始终相对于特定的旋转轴来定义。所有这些质量元素的总和称为对质量的积分。

Usually, the mass element dm will be expressed in terms of the geometry of the object, so that the integration can be carried out over the object as a whole (for example, over a long uniform rod).

通常，质量元素dm将用物体的几何形状来表示，从而使积分可以在整个物体上进行（例如，对一根长而均匀的杆进行积分）。

Having called this a general form, it is probably appropriate to point out that it is a general form only for axes which may be called "principal axes", a term which includes all axes of symmetry of objects. The concept of moment of inertia for general objects about arbitrary axes is a much more complicated subject. The moment of inertia in such cases takes the form of a mathematical tensor quantity which requires nine components to completely define it.

将此形式称为一般形式后，可能适当指出，这种形式仅适用于可能被称为‘主轴’的轴，这一术语包括物体的所有对称轴。对于一般物体关于任意轴的惯性矩概念是一个更为复杂的问题。在这种情况下，惯性矩的形式是一个数学张量量，需要九个分量来完全定义它。

Examples of integration to get moment of inertia. 积分计算转动惯量的例子	Straight rod 直杆
	Cylinder 圆柱体
	Sphere 球体

索引转动惯量概念

HyperPhysics ***** 力学 ***** 转动

R Nave