Name: mars spin under no torque
Uploaded: 2024-03-07T13:22:29.000Z
Duration: 1 min 15 s
Channel: JoVE

토크 없는 모션은 외부 토크가 작용하지 않는 강체의 움직임을 나타냅니다.
축대칭 물체를 생각해 보십시오, z축이 대칭 축이고, 질량 중심이 회전 참조 프레임의 원점에서 정의되어 있습니다. 여기서 x축과 y축에 대한 관성 모멘트는 동일합니다.
관성 프레임의 양의 Z축이 각운동량 벡터를 따라 있고 회전 프레임의 양의 z축과 각도 θ를 만들도록 정의된 관성 참조 프레임을 고려하십시오.
따라서 각운동량은 두 가지 방법으로 단위 벡터로 표현할 수 있으며, 단위 벡터의 구성 요소를 동일시함으로써 물체의 각속도에 대한 방정식이 도출됩니다.
또한 각 변위 측면에서 각속도를 작성하고 구성 요소를 다시 동일시하면 토크가 없는 축대칭 강체에 대한 운동 방정식이 제공됩니다.
여기서 물체의 각운동량, 세차운동 및 스핀은 각도 θ와 함께 운동 전반에 걸쳐 일정하게 유지됩니다.

토크 프리 모션

Torque Free Motion

토크 프리 모션은 외부 토크가 작용하지 않을 때 공간에서 강체의 움직임을 의미합니다. 이러한 유형의 움직임은 우주 공간과 같이 외부 힘이나 마찰이 없는 환경에서 관찰할 수 있습니다. 예를 들어, 우주에서 화성의 회전은 토크 없는 움직임입니다. 화성은 축대칭 물체입니다. 즉, z축으로 지정된 회전하는 대칭축이 있다는 의미입니다. 회전 기준 프레임은 화성의 질량 중심이 원점에 있도록 정의됩니다. 이러한 배열은 x축과 y축에 대한 관성 모멘트가 동일하도록 보장합니다.

화성의 움직임을 더 자세히 분석하기 위해 추가적인 관성 기준계가 사용됩니다. 이 관성계에서 양의 Z축은 화성의 각운동량 벡터와 정렬됩니다. 이는 회전 프레임의 양의 z축과 각도 세타(θ)를 형성합니다. 화성의 각운동량은 단위 벡터를 사용하여 두 가지 다른 방식으로 표현될 수 있습니다. 이러한 단위 벡터의 구성요소를 동일시함으로써 물체의 각속도에 대한 방정식을 도출할 수 있습니다. 더욱이, 각속도를 각 변위로 표현하고 구성 요소를 다시 한 번 동일시함으로써 화성에 살고 있는 토크가 없고 축대칭이며 견고한 물체의 운동 방정식에 도달합니다.

화성의 운동 전체에 걸쳐 화성의 각운동량, 세차, 회전 및 각도 θ는 일정하게 유지됩니다. 이러한 상수는 화성의 토크 없는 움직임의 흥미로운 역학을 드러냅니다.

토크 없는 모션은 외부 토크가 작용하지 않는 강체의 움직임을 나타냅니다.
z축이 있는 축대칭 객체를 고려하십시오.

Education

교육

JoVE Core

Mechanical Engineering

기계공학

Unit 1

제1단원

3-Dimensional Kinetics of a Rigid Body

강체의 3차원 운동역학

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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 of inertia about the x and y axes equal.

The motion is also studied in a separate inertial frame of reference. In that frame, the positive Z-axis points along Mars&amp;#39;s angular momentum vector. The Z-axis forms an angle theta (&amp;#952;) with the positive z-axis of the rotating frame. Mars&amp;#39;s angular momentum can then be written in two different ways using unit vectors.

By matching the components of those unit-vector forms, the angular velocity of the object can be found. The angular velocity can also be written in terms of angular displacement. When those components are matched again, the equation of motion for the torque-free, axisymmetric rigid object is obtained. During the motion, Mars&amp;#39;s angular momentum, precession, spin, and the angle theta remain constant.

Mars Spin Under No Torque

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 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...

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

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

Explore how Mars rotates as a torque-free, axisymmetric rigid body. Learn about angular momentum, precession, spin, and the angle theta in space.

Mars Spin Under No Torque | 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

Mars Spin Under No Torque

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