# Calculate Inertial tensor of 5000 points

• Yuriick
In summary, the conversation discusses how to calculate the inertia tensor for a sample of 5000 points from an ellipsoidal blob in 3D with some orientation. The formula for the center of mass is given, but it is uncertain how to find the inertia tensor without knowing the mass of each particle. A suggestion is made to scale out the unknown mass and compute I/m, given that the r_i values and center of mass can be determined from the initial data set.
Yuriick

## Homework Statement

I was given a sample of 5000 points from an ellipsoidal blob (in 3D) that has some orientation. Assuming that each point has equal mass, I was asked to calculate the inertia tensor of this blob, then ﬁnd the principal axes to determine the orientation of this blob.

## Homework Equations

R = $\frac{1}{M}$$\sum m_{i}r_{i}$
where M is the sum of the masses of every point
R is the center of mass

I = - $\sum m_{i} [r_{i} - R][r_{i} - R]$

## The Attempt at a Solution

Since all the mass for every particle is the same, the formula for center of mass becomes,

R = $\frac{1}{5000}$$\sum r_{i}$

however I'm not sure how to find the inertial tensor since I don't know the mass of each particle.

Any help would be much appreciated, thanks!

Perhaps you do not know the mass of each particle (in your line with the CM, you seem to assume that every mass is of mass "m", therefore, M = 5000*m). Why not scale out this unknown m, adn compute I/m? I bet you know all the r_i from your initial data set, and it seems like you can compute R. Is there a more complicated thing stopping you, even in light of this simple suggestion?

## 1. What is an inertial tensor?

An inertial tensor, also known as a moment of inertia tensor, is a mathematical representation of an object's resistance to changes in rotational motion. It describes how the mass of an object is distributed around its center of mass.

## 2. How do you calculate the inertial tensor of a set of points?

The inertial tensor can be calculated by summing the individual moment of inertia tensors of each point in the set. This involves multiplying the mass of each point by its distance from the center of mass squared and summing these values along the three axes of rotation.

## 3. Why is the inertial tensor important in physics and engineering?

The inertial tensor is important because it helps determine an object's rotational stability and the amount of energy required to rotate it. It is used to analyze the dynamics of rotating systems, such as in the design of spacecraft or understanding the motion of celestial bodies.

## 4. Can the inertial tensor change for a given object?

Yes, the inertial tensor can change if the distribution of mass within an object changes. For example, if a mass is added or removed from an object, the inertial tensor will also change. Additionally, the orientation of an object can affect its inertial tensor.

## 5. Are there any simplifications or assumptions made when calculating the inertial tensor of a set of points?

Yes, there are some simplifications and assumptions that are often made when calculating the inertial tensor. For example, the points are usually assumed to have a constant mass density and to be evenly distributed around the center of mass. In reality, this may not always be the case, but these simplifications allow for a more straightforward calculation of the inertial tensor.

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