Moment of Inertia: Kinetic Energy, Momentum & Conservation

[GeneralOJB]

I read that for a rotating body the kinetic energy $E_k = \sum \frac{1}{2}mv^2 = \frac{1}{2}{\omega}^2∑mr^2 = \frac{1}{2}I{\omega}^2$ where $I$ is the moment of inertia.

If we did the same thing for momentum then $P = ∑mv = \omega\sum mr$

So why is angular momentum $I\omega=\omega\sum mr^2$? Shouldn't the momentum just be the sum of the momentum of all the particles, like we did with kinetic energy?

Also why should I believe that this quantity $I\omega$ is conserved?

 

[adjacent]

Use LaTeX
$E_k = \sum\frac{1}{2}mv^2 = \frac{1}{2}\sum m(\omega r)^2 = \frac{1}{2}\omega^2\sum mr^2 = \frac{1}{2}I\omega^2$

 

[GeneralOJB]

Ah, didn't know we had LaTeX.

 

[A.T.]

Shouldn't the momentum just be the sum of the momentum of all the particles.

That would be the total linear momentum, not the total angular momentum.

Note that linear and rotational kinetic energy are both of the same physical scalar quantity. While linear and angular momentum are two different vector quantities. You should look at the vector formulas for momentum to understand it better.

 

[PJC01]

I would like to clarify that the formula for kinetic energy of a rotating body is correct, but the formula for momentum is not entirely accurate. The correct formula for angular momentum is actually $L=I\omega$, where $L$ is the angular momentum, $I$ is the moment of inertia, and $\omega$ is the angular velocity. This formula is derived from the definition of angular momentum as the cross product of position and linear momentum, and it takes into account the distribution of mass in the rotating body.

To address the question of why the momentum is not simply the sum of the momentum of all the particles, it is important to consider the concept of rotational inertia. Unlike linear motion, where the mass of an object is the only factor affecting its inertia, in rotational motion, the distribution of mass also plays a role. This is where the concept of moment of inertia comes in. It takes into account not only the mass of the object, but also the distance of each particle from the axis of rotation. This is why the formula for angular momentum includes the moment of inertia, as it reflects the rotational inertia of the object.

Furthermore, the conservation of angular momentum is a fundamental law of physics that has been observed and tested in countless experiments. It states that in a closed system, the total angular momentum remains constant, meaning it cannot be created or destroyed. This has been proven to be true in various scenarios, from simple spinning objects to more complex systems such as planetary orbits. Therefore, we have strong evidence to believe that the quantity $I\omega$, which represents the angular momentum, is conserved.

In conclusion, the formulas for kinetic energy and momentum in rotational motion take into account the distribution of mass and the rotational inertia of the object. The conservation of angular momentum is a well-established law of physics that has been observed and tested in various scenarios. As scientists, we must rely on evidence and observations to support our understanding of physical phenomena.