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

Considere un cuerpo rígido de masa 'm' y un centro de masa en el punto G, girando en un marco de referencia inercial.
En un punto arbitrario P, el momento angular se determina tomando el producto cruzado del vector de posición y el vector de momento lineal para cada elemento de masa.
La velocidad de un elemento de masa se compone de su velocidad de traslación y la velocidad relativa causada por la rotación del cuerpo.
Al sustituir la ecuación de velocidad en la ecuación del momento angular, expandir el producto cruzado e integrarlo en toda la masa se obtiene el momento angular total alrededor del punto P.
Aquí, si se elige el punto P como el centro de masa del cuerpo, entonces la primera integral se convierte en cero. Si se elige el punto P para que sea un punto fijo, entonces el término de velocidad lineal se desvanece.
Para cualquier otro punto arbitrario, la integral se puede simplificar. Aquí, el primer término da el momento debido al momento lineal, y el segundo término da el momento angular en el centro de masa del objeto.

momento angular respecto de cualquier punto arbitrario

Angular Momentum about an Arbitrary Axis

Imagine un cuerpo rígido con una masa denominada 'm', que tiene su centro de masa en el punto G y gira alrededor de un sistema de referencia inerte. El momento angular en un punto arbitrario P se puede calcular tomando el producto cruzado del vector de posición y el vector de momento lineal para cada elemento de masa individual.

La velocidad de un elemento de masa comprende su velocidad de traslación y la velocidad relativa instigada por la rotación del cuerpo. Sustituyendo la ecuación de velocidad en la ecuación del momento angular, expandiendo el producto cruz e integrando toda la masa se obtiene el momento angular total alrededor del punto P.

Si se selecciona el punto P como centro de masa del cuerpo, entonces la primera integral se vuelve cero a medida que el vector de posición se vuelve cero. Si se elige el punto P como punto fijo, entonces el término de velocidad lineal desaparece. Para cualquier otro punto arbitrario, la integral se puede simplificar. En este caso, el primer término proporciona el momento debido al momento lineal, mientras que el segundo término ofrece el momento angular en el centro de masa del objeto. Este enfoque proporciona una comprensión integral del momento angular en relación con puntos variables en un cuerpo en rotación.

Momento angular respecto de cualquier punto arbitrario

Considere un cuerpo rígido de masa 'm' y un centro de masa en el punto G, girando en un marco de referencia inercial.
En un punto arbitrario P, el ángulo ...

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JoVE Core

Mechanical Engineering

Ingeniería Mecánica

Unit 1

Unidad 1

3-Dimensional Kinetics of a Rigid Body

Cinética tridimensional de un cuerpo rígido

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

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

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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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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 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.

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Rigid Body Motion: Force and Moment Balance

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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 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.

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Angular Momentum Relative to Point P

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