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

一般的な平面運動、並進運動と回転運動の組み合わせを受ける剛体を考えてみましょう。
ニュートンの第二法則は、剛体の並進運動の方程式を与えます。それを時間間隔 dt と乗算し、積分の極限で積分すると、線形インパルスの原理の方程式が得られます。
線形インパルスの原理は、物体の運動量の変化が物体に加えられるインパルスに正比例することを明確に示しています。この方程式は、3 つの長方形のコンポーネントを使用して表すことができます。
一方、回転運動の方程式は、物体の重心付近の慣性モーメントとその角速度の積の時間微分として書くことができます。
ここで、慣性モーメントは一定であり、この方程式を積分形式に変換すると、回転インパルスの原理が得られます。
ここで、剛体の慣性モーメントと角速度の積は角運動量に等しく、角運動量は3つの長方形のコンポーネントを使用して表すことができます。

衝動と瞬間の原理

Principle of Impulse and Moment

本質的に並進運動と回転運動の混合である平面運動を受ける剛体を考えるとき、ニュートンの第 2 法則を適用すると、そのような体の並進運動の公式が得られます。 この方程式に時間間隔 dt を乗算し、積分限界を超えて積分すると、線形インパルスの原理を具体化した方程式が得られます。

<img alt="equation1" src="/files/ftp_upload/15550/15550_PT_equation_001.svg" style="height: 50px;">

ここで、点 G はオブジェクトの質量中心を表します。

線形力積の原理は、物体の運動量の変化がそれに加えられる力積に比例することを伝える概念です。 この特定の方程式は、3 つの長方形コンポーネントを使用して表すことができます。

対照的に、回転運動の方程式は、物体の質量中心の周りの慣性モーメントとその角速度の積の時間変化率 (時間微分値) として表されます。 この場合、慣性モーメントは一定のままです。 この方程式を積分形式に変換すると、回転力積の原理が得られます。

<img alt="equation2" src="/files/ftp_upload/15550/15550_PT_equation_002.svg" style="height: 50px;">

この原理では、剛体の慣性モーメントと角速度の積が角運動量に相当します。 この関係は、3 つの長方形コンポーネントを使用して説明できます。 したがって、この物語では、平面運動における剛体の複雑な力学を探求し、線形および回転衝撃の原理に焦点を当てます。

一般的な平面運動、並進運動と回転運動の組み合わせを受ける剛体を考えてみましょう。
ニュートンの第二法則は方程式を与えます...

Education

JoVE Core

Mechanical Engineering

機械工学

Unit 1

ユニット1

3-Dimensional Kinetics of a Rigid Body

剛体の三次元動力学

KEY TERMS AND CONCEPTS

主要用語および概念

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

impulse-and-angular-momentum-in-rigid-bodies

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

moment-of-inertia-about-an-arbitrary-axis

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

angular-momentum-and-principle-axes-of-inertia

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 機械工学は、平衡、力、トラス解析、構造力学などの重要なトピックをカバーする簡潔でわかりやすいアニメーション ビデオを提供します。実際のアプリケーションを探索し、科学者主導の実験を通じてこれらのコンセプトが実際に動作しているのを目撃してください。

mechanical engineering

Impulse and Angular Momentum in Rigid Bodies

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