Conservation of Angular momentum

[kyin01]

Homework Statement

Why is angular momentum conserved?

Homework Equations

$$\vec{\tau}= \vec{r} x \vec{F}$$
magnitude of $$\tau = I \alpha$$

The Attempt at a Solution

I first don't agree on why angular momentum is conserved, because if the radius change doesn't that change the torque?

And by definition angular momentum of a system is conserved when there are no external torque. While for particles, angular moment is conserved when there is no change in torque.

So the torque changes when radius is shortened, why is it that angular momentum is conserved?

 

[Dick]

r and F are parallel. They are both directed towards the hole. The angle between them is zero so Fxr vanishes, regardless of the magnitude of r changing.

 

[Moetasim]

Angular momentum is conserved because it is a fundamental principle of physics known as the law of conservation of angular momentum. This law states that the total angular momentum of a closed system remains constant over time, unless acted upon by an external torque. This means that even though the torque may change, the total angular momentum of the system will remain constant.

The reason for this conservation is due to the fact that angular momentum is a rotational analog of linear momentum and is a conserved quantity in nature. Just like how linear momentum is conserved due to Newton's third law of motion, angular momentum is conserved due to the law of conservation of angular momentum.

In the case of a changing radius, the torque may change but the moment of inertia (I) also changes in such a way that the angular velocity (ω) remains constant. This results in the conservation of angular momentum. This can be seen in the equation τ = Iα, where I is the moment of inertia and α is the angular acceleration. If the angular velocity remains constant, then the angular acceleration is zero and thus the torque (τ) is also zero, leading to the conservation of angular momentum.

In conclusion, angular momentum is conserved because it is a fundamental principle of physics and is a conserved quantity in nature. It is not dependent on changes in torque, but rather on the conservation of angular velocity and moment of inertia.