16.4
Momento angular respecto de cualquier punto arbitrario
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. E…
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.
View the full transcript and gain access to JoVE Core videos
Q1: How is angular momentum calculated at an arbitrary point on a rotating rigid body?
Angular momentum at an arbitrary point P is determined by taking the cross product of the position vector and linear momentum vector for each mass element. The velocity of each mass element combines its translational velocity and relative velocity from the body's rotation. Integrating these components over the entire mass yields the total angular momentum about point P.
Q2: What happens to the angular momentum equation when point P is at the center of mass?
When point P is selected as the center of mass of the body, the first integral in the angular momentum equation becomes zero because the position vector becomes zero. This simplification means the total angular momentum reduces to only the angular momentum at the center of mass, eliminating the moment due to linear momentum term.
Q3: How does choosing a fixed point simplify the angular momentum calculation?
If point P is chosen to be a fixed point in the inertial reference frame, the linear velocity term vanishes from the angular momentum equation. This simplification allows direct calculation of angular momentum without accounting for translational motion, making the analysis focus solely on rotational effects about that fixed point.
Q4: What are the two components of angular momentum for an arbitrary point not at the center of mass or fixed?
For any arbitrary point not at the center of mass or a fixed point, the angular momentum consists of two terms. The first term represents the moment due to linear momentum of the entire body, while the second term provides the angular momentum at the center of mass of the object.
Q5: Why is the velocity of a mass element composed of two components in a rotating rigid body?
In a rotating rigid body, each mass element experiences both translational motion of the body and rotational motion about the center of mass. The total velocity combines the translational velocity of the body and the relative velocity caused by the body's rotation, which is essential for accurately calculating angular momentum.
Q6: How does integrating over the entire mass contribute to finding total angular momentum?
By substituting the velocity equation into the angular momentum equation, expanding the cross product, and integrating over the entire mass, the contributions from all individual mass elements are combined. This integration process yields the total angular momentum about point P, accounting for the distributed mass throughout the rigid body.
Q7: What role does the inertial reference frame play in angular momentum calculations?
The inertial reference frame provides a non-accelerating coordinate system in which angular momentum is measured. All calculations of position vectors, velocities, and angular momentum for the rigid body are performed relative to this inertial frame, ensuring that the resulting angular momentum values are accurate and meaningful for rigid body dynamics analysis.