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

Si consideri un corpo rigido di massa 'm' e un centro di massa nel punto G, che ruotano in un sistema di riferimento inerziale.
In un punto arbitrario P, il momento angolare è determinato prendendo il prodotto incrociato del vettore posizione e del vettore momento lineare per ogni elemento di massa.
La velocità di un elemento di massa è composta dalla sua velocità di traslazione e dalla velocità relativa causata dalla rotazione del corpo.
Sostituendo l'equazione della velocità nell'equazione del momento angolare, espandendo il prodotto incrociato e integrando sull'intera massa si ottiene il momento angolare totale attorno al punto P.
Qui se si sceglie il punto P come centro di massa del corpo, allora il primo integrale diventa zero. Se il punto P viene scelto per essere un punto fisso, allora il termine di velocità lineare svanisce.
Per qualsiasi altro punto arbitrario, l'integrale può essere semplificato. Qui il primo termine dà il momento dovuto al momento lineare, e il secondo termine dà il momento angolare al centro di massa dell'oggetto.

momento angolare su qualsiasi punto arbitrario

Angular Momentum about an Arbitrary Axis

Immagina un corpo rigido con una massa indicata come "m", che ha il suo centro di massa nel punto G e ruota attorno a un sistema di riferimento inerziale. Il momento angolare in un punto arbitrario P può essere calcolato prendendo il prodotto incrociato del vettore posizione e del vettore momento lineare per ciascun singolo elemento di massa.

La velocità di un elemento di massa comprende la sua velocità di traslazione e la velocità relativa provocata dalla rotazione del corpo. Sostituendo l'equazione della velocità nell'equazione del momento angolare, espandendo il prodotto incrociato e integrando sull'intera massa, si ottiene il momento angolare totale attorno al punto P.

Se si seleziona il punto P come centro di massa del corpo, il primo integrale diventa zero poiché il vettore posizione diventa zero. Se si sceglie il punto P come punto fisso, il termine di velocità lineare scompare. Per qualsiasi altro punto arbitrario, l'integrale può essere semplificato. In questo caso il primo termine fornisce il momento dovuto al momento lineare, mentre il secondo termine offre il momento angolare al centro di massa dell'oggetto. Questo approccio fornisce una comprensione completa del momento angolare in relazione a diversi punti su un corpo rotante.

Momento angolare su qualsiasi punto arbitrario

Si consideri un corpo rigido di massa 'm' e un centro di massa nel punto G, che ruotano in un sistema di riferimento inerziale.
In un punto arbitrario P, l'angolare ...

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Mechanical Engineering

Ingegneria Meccanica

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3-Dimensional Kinetics of a Rigid Body

Cinetica tridimensionale di un corpo rigido

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TERMINI E CONCETTI CHIAVE

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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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 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

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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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