# Torque, angular momentum and a fixed axis-of-symmetry requirement

• rayoub
In summary, the conversation discusses the derivation of angular momentum for a rigid body around a fixed axis of rotation, which is shown to be L = Iω. However, it is mentioned that this only applies when the axis of rotation is an axis of symmetry. In contrast, the equation for torque around a fixed axis of rotation, τ = Iα, does not require the axis of symmetry assumption. The conversation also mentions the use of the "parallel axis theorem" to calculate moments of inertia about other axes, and the significance of the "principal moments of inertia" which are the axes where the object does not wobble when spun. The reason for the presentation in the book is not known.
rayoub
I'm reading through "University Physics 14th edition" by Young and Freeman. Section 10.5 on angular momentum for a rigid body around a fixed axis of rotation is derived as L = Iω. However, it shows that this is only the case for the fixed axis of rotation being an axis of symmetry.

In section 10.2 on torque it is shown that torque for a rigid body around a fixed axis of rotation is τ = Iα. However, in this case, it doesn't mention the need for an axis of symmetry for this to be true. I'm wondering if it is just an omission or if there is a specific reason why there is no need for an axis of symmetry assumption in this situation.

Thanks for any guidance.

I don't know the book but here is the deal:
1. Both Torque τ and Angular Momentum L can be defined and calculated about any arbitrary axis as can the moment of Inertia I
2. If you know I about the Center of Mass then it can easily be gotten for other axes using the "parallel axis theorem"
3. The "principal moments of Inertia" are the C of M moments about the symmetry axes and can be used to generate any other direction. These are the axes where the object does not want to "wobble"when spun.
Why the book presents them this way I don't know precisely.

Dale

## 1. What is torque?

Torque is a measure of the force that causes an object to rotate around an axis. It is calculated by multiplying the force applied to an object by the distance from the axis of rotation to the point where the force is applied.

## 2. What is angular momentum?

Angular momentum is a measure of the rotational motion of an object. It is calculated by multiplying the moment of inertia of the object by its angular velocity.

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

Torque and angular momentum are closely related, as torque is the force that causes a change in angular momentum. When a torque is applied to an object, it causes the object to rotate and therefore changes its angular momentum.

## 4. What is the fixed axis-of-symmetry requirement?

The fixed axis-of-symmetry requirement states that in order for an object to have a well-defined angular momentum, it must rotate around a fixed axis. This means that the axis of rotation does not change during the motion of the object.

## 5. How does the fixed axis-of-symmetry requirement affect the calculation of angular momentum?

The fixed axis-of-symmetry requirement simplifies the calculation of angular momentum, as it allows us to use a single axis to represent the rotational motion of the object. This makes it easier to calculate the moment of inertia and angular velocity, and ultimately determine the angular momentum of the object.

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