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

Consider a rigid body of mass &#39;m&#39; and a center of mass at point G, rotating in an inertial reference frame.
At an arbitrary point P, the angular momentum is determined by taking the cross product of the position vector and linear momentum vector for each mass element.
The velocity of a mass element is composed of its translational velocity and the relative velocity caused by the body&#39;s rotation.
By substituting the velocity equation into the angular momentum equation, expanding the cross product, and integrating over the entire mass gives the total angular momentum about point P.
Here if the point P is chosen as the center of mass of the body, then the first integral becomes zero. If the point P is chosen to be a fixed point, then the linear velocity term vanishes.
For any other arbitrary point, the integral can be simplified. Here the first term gives the moment due to linear momentum, and the second term gives the angular momentum at the center of mass of the object.

angular momentum about an arbitrary axis

Angular Momentum about an Arbitrary Axis

Imagine a rigid body with a mass denoted as &#39;m&#39;, which has its center of mass at point G and is rotating around an inertial reference frame. The angular momentum at an arbitrary point P can be calculated by taking the cross product of the position vector and linear momentum vector for each individual mass element.
The velocity of a mass element comprises its translational velocity and the relative velocity instigated by the body&#39;s rotation. Substituting the velocity equation into the angular momentum equation, expanding the cross product, and integrating over the entire mass yields the total angular momentum about point P.
If point P is selected as the center of mass of the body, then the first integral becomes zero as the position vector becomes zero. If point P is chosen to be a fixed point, then the linear velocity term disappears. For any other arbitrary point, the integral can be simplified. In this case, the first term provides the moment due to linear momentum, while the second term offers the angular momentum at the center of mass of the object. This approach provides a comprehensive understanding of the angular momentum in relation to varying points on a rotating body.

Consider a rigid body of mass &#39;m&#39; and a center of mass at point G, rotating in an inertial reference frame.
At an arbitrary point P, the angular ...

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3-Dimensional Kinetics of a Rigid Body

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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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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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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

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Angular Momentum Relative to Point P

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