# Moment of Inertia about an axis and Torque about a point

• I
• Nikhil_RG
In summary, Angular Momentum and Torque are defined about a point, while Moment of Inertia of a body is defined about an axis. There are equations that relate Angular Momentum and Torque with Moment of Inertia. In the case of a body free to rotate in any axis, the general equations of motion should be considered, without limitations on a specific axis. The cross product of the position vector of the particle and its linear momentum is used to calculate angular momentum, with a specific point chosen as the reference point. The Moment of Inertia Tensor is a second-order tensor and can be reduced to a scalar by restricting the rotation to a specific axis. However, in the case of no limitations, the relevant component of torque is the only one

#### Nikhil_RG

Angular Momentum and Torque are defined about a point. But Moment of Inertia of a body is defined about an axis. There are equations which connect Angular momentum and Torque with Moment of Inertia. How will this be consistent? When I say that the torque of a force acting on a body about a point causes it to rotate about an axis, which axis should be considered that includes the point about which the torque is acting.

Nikhil_RG said:
Angular Momentum and Torque are defined about a point. But Moment of Inertia of a body is defined about an axis.
According to whom?

Wikipedia:
Angular momentum is a vector quantity (more precisely, a pseudovector) that represents the product of a body's rotational inertia and rotational velocity (in radians/sec) about a particular axis.

Torque is defined as the product of the magnitude of the force and the perpendicular distance of the line of action of a force from the axis of rotation.

Moment of inertia is not defined relative to an axis. It is defined relative to a point. However, it is an order 2 tensor and not a scalar. In order to obtain a scalar, you can restrict the rotation of a body to only be possible around a particular axis. In this case, only the torque’s component in the axis direction will be relevant and angular momentum will be parallel to the axis.

vanhees71
Thank you malawi_glenn for the response.

My question comes from the fact that the basic expression to calculate angular momentum involves finding the cross product of the position vector of the particle and it's linear momentum. So there has to be a point about which the position vector is defined and the angular momentum would be calculated about that particular point.

Orodruin , is there a textbook or resource that I could refer to to understand about Moment of Inertia Tensor.

And in the case where the body is free to rotate in any axis and a force is acting at some point on it, which causes a Torque, which axis do we consider, since there are no limitations.

Nikhil_RG said:
Orodruin , is there a textbook or resource that I could refer to to understand about Moment of Inertia Tensor.
Well, first of all, do you understand tensors?

Nikhil_RG said:
And in the case where the body is free to rotate in any axis and a force is acting at some point on it, which causes a Torque, which axis do we consider, since there are no limitations.
You do not consider an axis. You consider the general equations of motion. There are some simplified cases such as an object rotating freely around a fixed point an object not subject to any net force (just torques).

Nikhil_RG said:
So there has to be a point about which the position vector is defined and the angular momentum would be calculated about that particular point.
In the case where you fix the rotational axis, only the torque in the axis’ direction is relevant. This component will not depend on which reference point you pick as long as you pick a point on the axis.

vanhees71

## 1. What is moment of inertia and how is it related to rotational motion?

Moment of inertia is a measure of an object's resistance to changes in its rotational motion. It is often defined as the sum of the products of each particle's mass and the square of its distance from the axis of rotation. In simpler terms, it describes how difficult it is to change an object's rotation.

## 2. How is the moment of inertia calculated for different shapes?

The moment of inertia for a point mass can be calculated using the formula I = mr², where m is the mass and r is the distance from the axis of rotation. For more complex shapes, the moment of inertia can be calculated using integration or by using tables of known values for common shapes such as cylinders, spheres, and discs.

## 3. What is the relationship between moment of inertia and torque?

Torque is the measure of the force that causes an object to rotate. It is directly proportional to the moment of inertia and the angular acceleration of the object. In other words, the larger the moment of inertia, the more torque is needed to rotate the object.

## 4. How does the moment of inertia affect an object's stability?

The moment of inertia plays a crucial role in an object's stability. Objects with a larger moment of inertia are more resistant to changes in their rotational motion, making them more stable. This is why objects with a low center of mass, such as a pyramid, are more stable than objects with a high center of mass, such as a pencil.

## 5. Can the moment of inertia be changed?

Yes, the moment of inertia can be changed by altering the mass distribution of an object. For example, a figure skater can increase or decrease their moment of inertia by extending or pulling in their arms, respectively. This change in moment of inertia affects their rotational speed and allows them to perform different spins and movements on the ice.