Name: angular momentum relative to point p
Uploaded: 2024-03-07T13:22:28.000Z
Duration: 1 min 11 s
Channel: JoVE

'm' kütleli katı bir cisim ve G noktasında eylemsiz bir referans çerçevesinde dönen bir kütle merkezi düşünün.
Keyfi bir P noktasında, açısal momentum, her bir kütle elemanı için konum vektörü ile doğrusal momentum vektörünün çapraz çarpımı alınarak belirlenir.
Bir kütle elemanının hızı, öteleme hızından ve cismin dönüşünün neden olduğu bağıl hızdan oluşur.
Hız denklemini açısal momentum denklemine koyarak, çapraz çarpımı genişleterek ve tüm kütle üzerine entegre ederek, P noktası etrafındaki toplam açısal momentumu verir.
Burada P noktası cismin kütle merkezi olarak seçilirse, ilk integral sıfır olur. P noktası sabit bir nokta olarak seçilirse, doğrusal hız terimi kaybolur.
Diğer herhangi bir keyfi nokta için, integral basitleştirilebilir. Burada ilk terim doğrusal momentumdan kaynaklanan momenti verir ve ikinci terim cismin kütle merkezindeki açısal momentumu verir.

herhangi bir rasgele noktaya göre açısal momentum

Angular Momentum about an Arbitrary Axis

Kütle merkezi G noktasında olan ve eylemsiz bir referans çerçevesi etrafında dönen, kütlesi "m" ile gösterilen katı bir cisim düşünün. Herhangi bir P noktasındaki açısal momentum, her bir kütle elemanı için konum vektörü ile doğrusal momentum vektörünün çapraz çarpımı alınarak hesaplanabilir.


 Bir kütle elemanının hızı, onun öteleme hızını ve cismin dönüşüyle tetiklenen bağıl hızı içerir. Hız denklemini açısal momentum denkleminde yerine koymak, çapraz çarpımı genişletmek ve tüm kütle üzerinde integral almak, P noktası etrafındaki toplam açısal momentumu verir.


 Eğer P noktası cismin kütle merkezi olarak seçilirse, konum vektörü sıfır olurken birinci integral de sıfır olur. P noktası sabit bir nokta olarak seçilirse doğrusal hız terimi kaybolur. Başka herhangi bir keyfi nokta için integral basitleştirilebilir. Bu durumda birinci terim doğrusal momentumdan kaynaklanan momenti verirken, ikinci terim cismin kütle merkezindeki açısal momentumu sunar. Bu yaklaşım, dönen bir cisim üzerindeki değişen noktalara göre açısal momentumun kapsamlı bir şekilde anlaşılmasını sağlar.

Herhangi Bir Rasgele Noktaya Göre Açısal Momentum

'm' kütleli katı bir cisim ve G noktasında eylemsiz bir referans çerçevesinde dönen bir kütle merkezi düşünün.
Keyfi bir P noktasında, açısal ...

Education

JoVE Core

Mechanical Engineering

Makine Mühendisliği

Unit 1

Ünite 1

3-Dimensional Kinetics of a Rigid Body

Katı Bir Cismin Üç Boyutlu Kinetiği

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

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 by rotation. Substituting this velocity into the angular momentum equation, expanding the cross product, and integrating over the entire mass gives the total angular momentum about point P.

The choice of point P changes the result in useful ways. If P is the center of mass, the first integral becomes zero because the position vector is zero. If P is a fixed point, the linear velocity term disappears. For any other arbitrary point, the integral can still be simplified. In that case, the first term gives the moment from linear momentum, and the second term gives the angular momentum at the center of mass of the object.

Angular Momentum Relative to Point P

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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.

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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...

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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 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...

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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.

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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...

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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.

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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 angular momentum is calculated about an arbitrary point in a rigid body, and how the center of mass or a fixed point changes the result.

Angular Momentum Relative to Point P | JoVE K12

JoVE Core Makine Mühendisliği, denge, kuvvetler, kafes kiriş analizi ve yapısal mekanik gibi temel konuları kapsayan kısa, anlaşılması kolay animasyonlu videolar sunar. Gerçek dünyadaki uygulamalarını keşfedin ve bilim insanlarının önderlik ettiği deneylerle bu kavramların uygulanmasına tanık olun.

mechanical engineering

Angular Momentum Relative to Point P

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