16.4
임의의 점에 대한 각운동량
질량 중심이 G 지점에 있고 관성 기준계를 중심으로 회전하는 'm'이라는 질량을 가진 강체를 상상해 보세요. 임의의 점 P에서의 각운동량은 각 개별 질량 요소에 대한 위치 벡터와 선형 운동량 벡터의 외적을 취하여 계산할 수 있습니다.
질량 요소의 속도는 병진 속도와 몸…
질량 'm'의 강체와 G 지점에서 질량 중심이 관성 참조 프레임에서 회전하는 것을 고려하십시오.
임의의 점 P에서 각운동량은 각 질량 요소에 대한 위치 벡터와 선형 운동량 벡터의 외적을 취하여 결정됩니다.
질량 요소의 속도는 평행 이동 속도와 물체의 회전으로 인한 상대 속도로 구성됩니다.
속도 방정식을 각운동량 방정식에 대입하고, 외적을 확장하고, 전체 질량에 걸쳐 적분하면 점 P에 대한 총 각운동량이 제공됩니다.
여기서 점 P를 물체의 질량 중심으로 선택하면 첫 번째 적분은 0이 됩니다. 점 P를 고정 점으로 선택하면 선형 속도 항이 사라집니다.
다른 임의의 점의 경우 적분을 단순화 할 수 있습니다. 여기서 첫 번째 항은 선형 운동량으로 인한 모멘트를 제공하고 두 번째 항은 물체의 질량 중심에서 각운동량을 제공합니다.
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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.