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

トルクフリーモーションとは、外部トルクが作用しない剛体の動きを指します。
軸対称オブジェクトについて考えてみて、z 軸を対称軸とし、回転する基準座標系の原点に重心を定義します。ここで、x 軸と y 軸の周りの慣性モーメントは等しくなります。
慣性座標系の正の Z 軸が角運動量ベクトルに沿っており、回転座標系の正の Z 軸と角度 θ になるように定義された慣性座標系について考えます。
したがって、角運動量は2つの方法で単位ベクトルで表すことができ、単位ベクトルの成分を等しくすることで、オブジェクトの角速度の方程式が導出されます。
さらに、角速度を角変位の観点から記述し、コンポーネントを再度等価にすると、トルクのない軸対称剛体の運動方程式が得られます。
ここでは、物体の角運動量、歳差運動量、およびスピンは、角度θとともに運動全体を通じて一定のままです。

トルクの無い運動

Torque Free Motion

トルクのない運動とは、外部トルクが作用していないときの空間内の剛体の動きを指します。 この動きは、宇宙空間のように外力や摩擦がない環境で観察できます。 たとえば、宇宙での火星の回転はトルクのない運動です。 火星は軸対称の天体です。つまり、それに沿って回転する対称軸 (z 軸と呼ばれます) があることを意味します。 回転座標系は、火星の質量中心が原点にあるように定義されます。 この配置により、x 軸と y 軸の周りの慣性モーメントが確実に等しくなります。

火星の動きをさらに分析するために、追加の慣性座標系が使用されます。 この慣性系では、正の Z 軸は火星の角運動量ベクトルと一致します。 回転フレームの正の Z 軸と角度シータ (θ) を形成します。 火星の角運動量は、単位ベクトルを使用して 2 つの異なる方法で表現できます。 これらの単位ベクトルの成分を等式化することにより、物体の角速度の方程式を導き出すことができます。 さらに、角速度を角変位で表し、その成分を再度等式化することにより、火星に住むトルクのない軸対称の剛体の運動方程式に到達します。

火星の運動全体を通して、その角運動量、歳差運動、自転、および角度 θ は一定のままです。 これらの定数は、火星のトルクの無い運動の興味深い力学を明らかにします。

トルクフリーモーションとは、外部トルクが作用しない剛体の動きを指します。
軸対称オブジェクトを考えてみて、z軸...

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

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

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mechanical engineering

Mars Spin Under No Torque

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