# Angular Momentum: Conditions & Answers

• enippeas
In summary, the formula L = I ω for angular momentum can be applied around any axis, but the inertia momentum I will be different depending on the direction of the rotation. The value of I is affected by the distribution of an object's mass relative to the rotation axis, with a smaller I making rotation easier. This can be seen in the example of ballet dancers raising their arms to spin faster. The parallel axis theorem states that the Moment of Inertia about an axis can be calculated by adding the Moment of Inertia about the parallel axis through the center of mass and the product of the mass and the square of the distance from the axis to the center of mass.

#### enippeas

First of all, sorry for my poor english
I am studying classical mechanics and have a problem with angular momentum. I am looking many books but i can't find an exactly answer. Under what conditions can i apply L = I ω; So far i understand that i can apply this formula around either the cm or a fixed point. Is that correct; Is there any other point; Thanks

That is the definition of angular momentum for an object.

You only need to choose which is the REFERENCE AXIS for the rotations considered, because the inertia momentum I is not simply a scalar quantity, but is more complicated than that. Technically it's a tensor. It means that the value of I is different depending on which directions you are looking at it.

As a help, notice the great similarity between L=Iw (angular momentum) and p=mv (momentum). They are indeed the same thing, except that the first is for rotations and the second for traslations.

However, while m (mass) is just one simple value that works whatever the direction of the movement, I is not as simple.

Hint: think of an object with a long shape, such a wooden pole. Trying to make it spin requires an easier effort if you'r spinning it around its own axis rather than trasversally. That is represented by the fact that I' (referring to the pole's axis) is smaller than I'' (referring to an axis perpendicular to the pole and crossing the pole's middle point). If you try to rotate the pole horizontally and by holding one extremity rather than the middle point, it's even more difficult: I''' (ref to an axis perpendicular to the pole but crossing one of the extremity) is even greater.

To get the same rotational speed, you need a bigger effort L when I is greater.

Conversely, with equal effort L you speed up more an object that has a small I.

I depends on how the object's mass is distributed, relative to the rotation axis. The closer the mass is located, or "lumped" around the axis, the smaller I and hence the "easier" the rotation. Typical example of this is the ballet dancers, who first spin themselves with arms extended, then they raise their arms above their head (this way, they move part of their body mass closer to the rotational axis) and get a faster rotation.

Welcome to PF!

enippeas said:
Under what conditions can i apply L = I ω; So far i understand that i can apply this formula around either the cm or a fixed point. Is that correct; Is there any other point; Thanks

Hi enippeas ! Welcome to PF!

L = I ω is valid about any axis.

But of course we must use the I for that axis, which will always be larger than the I for the parallel axis through the c.o.m.

From the PF Library:
The parallel axis theorem:

The Moment of Inertia of a body about an axis is

$$I = (I_C\,+\,md^2)$$

where m is the mass, d is the distance from that axis to the centre of mass, and $I_C$ is the Moment of Inertia about the parallel axis through the centre of mass.

(oh, and it's "I have looked in many books, but I couldn't find an exact answer." )

## 1. What is angular momentum?

Angular momentum is a physical quantity that represents the rotational motion of an object. It is defined as the product of an object's moment of inertia and its angular velocity.

## 2. What are the conditions for conservation of angular momentum?

The conditions for conservation of angular momentum are: there must be no external torque acting on the system, the system must be isolated, and the moment of inertia of the system must remain constant.

## 3. How is angular momentum related to torque?

Angular momentum and torque are related through the equation L = Iω, where L is angular momentum, I is moment of inertia, and ω is angular velocity. This means that an increase in torque will result in an increase in angular momentum, and vice versa.

## 4. Is angular momentum a vector or scalar quantity?

Angular momentum is a vector quantity because it has both magnitude and direction. The direction of angular momentum is perpendicular to the plane of rotation.

## 5. How is angular momentum conserved in a closed system?

In a closed system, angular momentum is conserved because there is no external torque acting on the system. This means that the total angular momentum of the system remains constant, even if individual objects within the system may experience changes in angular momentum.