16.4
Herhangi Bir Rasgele Noktaya Göre Açısal Momentum
Kütle merkezi G noktasında olan ve eylemsiz bir referans çerçevesi etrafında dönen, kütlesi "m" ile gösterilen katı bir cisim düşünün. Herhangi bir P…
'm' kütleli katı bir cisim ve G noktasında eylemsiz bir referans çerçevesinde dönen bir kütle merkezi düşünün.
Keyfi bir P noktasında, açısal momentum, her bir kütle elemanı için konum vektörü ile doğrusal momentum vektörünün çapraz çarpımı alınarak belirlenir.
Bir kütle elemanının hızı, öteleme hızından ve cismin dönüşünün neden olduğu bağıl hızdan oluşur.
Hız denklemini açısal momentum denklemine koyarak, çapraz çarpımı genişleterek ve tüm kütle üzerine entegre ederek, P noktası etrafındaki toplam açısal momentumu verir.
Burada P noktası cismin kütle merkezi olarak seçilirse, ilk integral sıfır olur. P noktası sabit bir nokta olarak seçilirse, doğrusal hız terimi kaybolur.
Diğer herhangi bir keyfi nokta için, integral basitleştirilebilir. Burada ilk terim doğrusal momentumdan kaynaklanan momenti verir ve ikinci terim cismin kütle merkezindeki açısal momentumu verir.
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.