# What is the Correct Calculation for the Inertia Tensor in a 2D Rotation?

In summary: I are calculated using the following equation:I_{xx} = m_i y_{i}^{2}I_{yy} = m_i x_{i}^{2}I_{xy} = I_{yx} = - m_i x_i y_i
my homework problem deals w/ rotation in x-y plane. so the tensor is only 2d. inertial tensor still seems obscure to me... my question for now is purely mathematical. assuming the basis are (x,y). I calculated the components of I, is the following correct?
$$I_{xx} = m_i y_{i}^{2}$$
$$I_{yy} = m_i x_{i}^{2}$$
$$I_{xy} = I_{yx} = - m_i x_i y_i$$

for some reason my professor wrote...
$$I_{xx} = m_i x_{i}^{2}$$
and I'm pretty sure its not right

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You're fine (besides the missing summation symbols). Think of it this way: if you are rotating about the x-axis, would your inertia depend on where you are along the x-axis or how far away you are from the x-axis.

what about the off diagonal terms? are they moments about x=y line?... also do you need orthogonal basis for calculating inertial tensor? from the general definition it doesn't look like a requirement.

Sorry, I don't have a good analogy in mind for the products of inertia; maybe someone else reading the thread can help us out. I guess it comes into play when you have rotation that is not along one of the principal axes.
I suppose you could calculate the inertial tensor using a different basis, but wouldn't that just complicate the algebra(and unnecessarily at that)? I'm sorry if I haven't been much help.

The full tensor (diagonal and off diagonal terms) is important for rotations about any axis through the origin in the x.y plane, not just the y = x line. The tensor product of I with the angular velocity vector gives the angular momentum vector. When the angular velocity vector is along the x or y axes, the off diagonal terms do not contribute.

I can see why angular momentum is in same direction as angular velocity if object rotates about principal axes only. My textbook derives the inertia tensor through rotational kinetic energy calculation. I can see that it relates energy w/ velocity... but i don't understand in what way products of inertia describe the relationship... perhaps i just need to do some problems and see how it all works out.

## What is an inertial tensor?

An inertial tensor is a mathematical representation of the distribution of mass and moments of inertia of a rigid body. It is used to describe how an object will respond to rotational forces.

## Why is the calculation of inertial tensor important?

The inertial tensor is important in understanding the dynamics and stability of a rigid body in motion. It is used in various fields such as engineering, physics, and robotics to predict and control the behavior of objects.

## How is the inertial tensor calculated?

The inertial tensor can be calculated using the mass distribution and geometry of the object. It involves integrating the mass distribution over the object's volume and multiplying it by the distance of the mass elements from the object's axis of rotation.

## What factors can affect the calculation of the inertial tensor?

The shape and mass distribution of the object are the main factors that affect the calculation of the inertial tensor. Other factors such as the object's orientation and the choice of reference frame can also have an impact.

## How is the inertial tensor used in real-life applications?

The inertial tensor is used in various real-life applications such as spacecraft and aircraft design, robotics, and video game physics. It is also used in sports equipment design, such as tennis rackets and golf clubs, to optimize performance and stability.

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