Integration in angular momentum

The mass of the pancake is not changing, so the integral is constant. In summary, the conversation discusses the expression of \mathbf{L}=\int \mathbf{r} \times \mathbf{\pi}(\mathbf{r}) d^3\mathbf{r}, where \mathbf{r} is the position vector, \mathbf{\pi}(r) is the momentum density, and \rho(r) is the mass density. The question is raised about the use of dm in this expression, and it is clarified that dm = \rho(\mathbf{r}) d^3\mathbf{r}. The conclusion is that the integral over the mass at some instant is constant
  • #1
Rikudo
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Homework Statement
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Relevant Equations
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https://www.physicsforums.com/threa...f-a-translating-and-rotating-pancake.1005990/
So,I think I posted this in the wrong place. So, I will move it to here.
Here, in post #6, it is stated that ##\int R dm = M R##. As far as I know, R change from time to time and it is not constant. Hence, isn't it incorrect to say that ##\int R dm = M R##? Or, are there any techniques that are skipped?
 
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  • #2
Though it is not an answer to your question but a matter of taste, I am not familiar with expression dm. I prefer to say it
[tex]\mathbf{L}=\int \mathbf{r} \times \mathbf{\pi}(\mathbf{r}) d^3\mathbf{r} =\int \mathbf{r} \times \rho(\mathbf{r}) \mathbf{v}(\mathbf{r}) d^3\mathbf{r} [/tex]
where ##\pi(r)## is momentum density and ##\rho(r)## is mass density.
So now I know
[tex]dm = \rho(\mathbf{r}) d^3\mathbf{r}[/tex]
 
  • #3
Rikudo said:
Here, in post #6, it is stated that ##\int R dm = M R##. As far as I know, R change from time to time and it is not constant.
Right, but that integral has no moving parts. It is the integral over the mass at some instant.
 

Related to Integration in angular momentum

1. What is angular momentum?

Angular momentum is a physical quantity that describes the rotational motion of an object. It is a vector quantity that takes into account both the mass and velocity of an object as well as its distance from a fixed point.

2. How is angular momentum calculated?

Angular momentum is calculated by multiplying the moment of inertia (a measure of how an object's mass is distributed around its axis of rotation) by the angular velocity (the rate of change of an object's angular position).

3. What is the principle of conservation of angular momentum?

The principle of conservation of angular momentum states that in a closed system, the total angular momentum remains constant. This means that if no external torque is applied, the angular momentum of the system will remain the same.

4. How is angular momentum related to rotational inertia?

Rotational inertia, or moment of inertia, is a measure of an object's resistance to changes in its rotational motion. It is directly proportional to an object's angular momentum, meaning that the greater the rotational inertia, the greater the angular momentum of the object.

5. What is the role of angular momentum in physics?

Angular momentum plays a crucial role in many areas of physics, including classical mechanics, quantum mechanics, and astrophysics. It helps us understand the rotational motion of objects, the stability of rotating systems, and the behavior of celestial bodies such as planets and stars.

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