Name: impulse and angular momentum in rigid bodies
Uploaded: 2025-05-22T09:06:17.000Z
Duration: 1 min 15 s
Channel: JoVE

考虑一个刚体正在进行一般的平面运动，即平动和转动的组合。
牛顿第二定律给出了刚体平动运动的方程。将该方程乘以时间间隔 dt 并对积分限进行积分，可得到线性冲量原理的方程。
线性冲量原理指出，物体动量的变化量与作用在其上的冲量成正比。该方程可用三个矩形分量表示。
另一方面，转动运动的方程可以写成物体关于其质心的转动惯量与其角速度乘积对时间的导数。
此处，转动惯量为常量；将此方程转化为积分形式后，即得到旋转冲量原理。
此处，刚体的转动惯量与角速度的乘积等于角动量，该角动量可用三个矩形分量表示。

冲量和力矩原理

Principle of Impulse and Moment

当人们思考一个刚体所进行的平面运动（本质上是平移运动和旋转运动的混合）时，牛顿第二定律的应用给出了这种物体平移运动的公式。如果将该方程与时间间隔dt进行相乘，然后在其极限上进行积分，则得到体现线性脉冲原理的方程。

<img alt="Equation 1" src="/files/ftp_upload/15550/15550_PT_equation_001.svg" style="text-align: center; height: 50px;">
这里的下标G代表了物体的质心。

线性冲量的原理是一个概念，它传达了物体动量的变化与施加到其中的冲量成正比。这个特定的方程可以通过使用三个矩形分量来进行表示。

相反的，旋转运动方程表示为围绕物体质心的惯性矩与其角速度的乘积随时间的变化率（时间导数）。在这种情况下，转动惯量保持其恒定状态。如果将这个方程转化为积分形式，便能够得出旋转冲量原理。

<img alt="Equation 2" src="/files/ftp_upload/15550/15550_PT_equation_002.svg" style="text-align: center; height: 50px;">
在这个原理中，刚体的转动惯量和角速度的乘积等于角动量。这种关系可以使用三个矩形分量来进行说明。因此，这个叙述探讨了平面运动中刚体的复杂动力学，并强调了其线性和旋转脉冲的原理。

当考虑一个进行平面运动的刚体时，这种运动本质上是平动与转动的结合，其应用涉及……

Education

教育

JoVE Core

Mechanical Engineering

机械工程

Unit 1

第一单元

3-Dimensional Kinetics of a Rigid Body

刚体的三维动力学

KEY TERMS AND CONCEPTS

关键术语与概念

impulse and angular momentum in rigid bodies

Impulse and angular momentum help describe a rigid body moving in a plane. This motion combines translation and rotation. The topic uses Newton's second law to connect force, momentum, and motion.

For translational motion, the equation is written for the center of mass, labeled G. When this equation is multiplied by a small time interval, dt, and integrated, it gives the principle of linear impulse. This principle shows that a change in momentum is proportional to the impulse applied to the body. The result can also be written in three rectangular components.

Rotational motion is treated in a similar way. The time rate of change of the product of the moment of inertia about the center of mass and the angular velocity gives the rotational equation. Here, the moment of inertia is constant. When this form is written as an integral, it becomes the principle of rotational impulse.

In the rotational case, the product of the moment of inertia and angular velocity is the angular momentum. This relationship can also be shown with three rectangular components. Together, these ideas describe the linear and rotational impulse behavior of a rigid body in plane motion.

Impulse and Angular Momentum in Rigid Bodies

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moments and product of inertia

Moment and Product of Inertia Basics

Moment of inertia and product of inertia describe how mass is distributed in a rigid body. They help show how the body responds about coordinate axes and planes. For a small mass element, the moment of inertia is found by multiplying the element's mass by the square of the shortest distance to one of the three coordinate axes. That same idea is extended to the whole body by integration. Adding the contributions of all the small mass elements gives the body's moment of inertia about that axis.

inertia tensor

Principal Axes and Moments of Inertia

The inertia tensor describes how mass is spread out in a rigid body and how that body resists rotation. It is written as a 3×3 matrix, and each entry represents a moment of inertia about a particular axis. This makes the tensor a compact way to study rotational behavior in three dimensions. The diagonal entries show the moments of inertia about the principal axes. These principal axes are the directions where the object has the least resistance to rotation. A smaller moment of inertia along...

moment of inertia about an arbitrary axis

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Calculating Moment of Inertia on Any Axis

Moment of inertia can be calculated about any chosen axis, not just a principal axis. It depends on the mass distribution of the object along that axis. This makes it useful when an object rotates in a real machine or other setup where balance and stability matter. To find the moment of inertia about an arbitrary axis, the perpendicular distance from the axis to each mass element is needed. That distance is obtained with a cross product between the unit vector that gives the axis direction and...

angular momentum about an arbitrary axis

Angular Momentum Relative to Point P

Angular momentum relative to point P is found by adding the motion of every mass element in a rigid body. The body has mass m and a center of mass at point G. It rotates in an inertial reference frame, and the angular momentum at P comes from the cross product of each position vector and its linear momentum. To build the full expression, the velocity of each mass element is written as two parts. One part is the translational velocity of the body. The other part is the relative velocity caused...

angular momentum and principle axes of inertia

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Angular Momentum in Principal Axes

Angular momentum in a rigid body can be written using rectangular coordinates. The body’s angular momentum comes from the cross-product of each mass element’s position vector with the cross-product of the body’s angular velocity and that position vector. This gives a compact way to describe how a solid structure moves in rotation. To work with the equation more easily, the XYZ axes can be chosen as a new set of rectangular axes. These axes may be inclined at any angle to the reference frame.

principle of impulse and moment

Impulse and angular momentum help describe a rigid body moving in a plane. This motion combines translation and rotation. The topic uses Newton's second law to connect force, momentum, and motion. For translational motion, the equation is written for the center of mass, labeled G. When this equation is multiplied by a small time interval, dt, and integrated, it gives the principle of linear impulse. This principle shows that a change in momentum is proportional to the impulse applied to the...

kinetic energy for a rigid body

Rigid Body Energy from Center and Rotation

Rigid body kinetic energy in planar motion can be written using the body’s center of mass and its rotation. The solid object moves in general planar motion, and its center of mass is marked at point G. To find the kinetic energy of the i-th particle, the relative velocity definition is used with the position vector rA from point A to that particle. The kinetic energy of the whole body is built from the particles. First, a scalar product is taken. Then the expression is written in integral form.

equation of motion for a rigid body

Rigid Body Motion: Force and Moment Balance

Rigid body motion is described with both translational motion and rotational motion about the center of mass, point G. The center of mass is the point where Newton's Second Law applies to translation. It gives the linear motion of the body. The same body also has rotational motion about point G. The combined moments about the center of mass equal the rate of change of angular momentum. A moment is the turning effect caused by a force, and an external force applied away from the center of mass...

euler equations of motion

Rigid Body Rotation in Euler’s Frame

Euler’s equations of motion describe how a rigid body rotates when measured in a moving frame attached to the body. The body rotates with angular velocity ω in an inertial frame, and the attached frame rotates with angular velocity Ω. Together, these two frames help describe the motion of the rigid body around its center of mass. The total moment about the center of mass is found by adding two parts. One part is the rate of change of angular momentum about the center of mass in the rotating...

torque free motion

Mars Spin Under No Torque

Torque-free motion describes the movement of a rigid body when no external torques act on it. It can be seen in places with no outside forces or friction, such as outer space. Mars is one example of this kind of motion because it rotates in space without external torque. Mars is an axisymmetric object, which means it has an axis of symmetry. That symmetry defines the rotating z-axis. The rotating frame is set so that Mars&amp;#39;s center of mass is at the origin. This setup makes the moments...

key terms and concepts

Learn how linear impulse and rotational impulse describe rigid body motion in a plane, including center of mass, angular velocity, and angular momentum.

Impulse and Angular Momentum in Rigid Bodies | JoVE K12

JoVE Core Mechanical Engineering offers concise, easy-to-understand animated videos covering key topics such as equilibrium, forces, truss analysis, and structural mechanics. Explore their real-world applications, and witness these concepts in action through scientist-led experiments.

mechanical engineering

Impulse and Angular Momentum in Rigid Bodies

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