# Find Moment of Inertia Around CoM: Summation Formula & Point Mass

• genericpumpkin
In summary, the conversation discusses how to find the moment of inertia around the center of mass of an object when the axis of rotation is not through the center of mass. The parallel axis theorem is mentioned as a method to transfer the moment of inertia from the center of mass to the axis of rotation. The conversation also touches on the use of summation formulas and clarifies the variables used in the equation for calculating moment of inertia. The final output of the conversation is a simplified equation for a uniform density rectangular area moment.
genericpumpkin
How do I find the moment of inertia around the CoM of an object when the axis of rotation is not through the CoM?

When Are summation formula used in equations and what exactly constitutes a point mass? regarding moments of inertia?

Use the parallel axis theorem to transfer the MOI from the c.o.m. to the axis of rotation.

I took the concept of a parallel axis shift and didn't require it to be parallel.

DoubleIntegral from -r_y to(L_y-r_y) and -r_x to (L_x - r_x) of ((x-r_x)^2+(y-r_y)^2)dm
For dm I used Mdxdy/(L_x)(L_y)

Clarifying, x and y are my variables, L_x is the x length of my object, and L_y is the length in the y direction. r_x and r_y are the distances or "shifts" of my axis from the center of mass in the x and y directions respectively.

Here's what I got (after quite a bit of simplification)

M [ (L_x^2)/3 - 2(L_x)(r_x) +3r_x^2 + (L_y^2)/3 - 2(L_y)(r_y) + 3r_y^2]

Anyone care to check my work? Or at least my initial setup? I didn't really get as much out of moments as I could've in my mechanics class, so I took this opportunity to brush up on them.
Btw I'm integrating a uniform density rectangular area moment. I hope you followed that lol

## 1. What is the moment of inertia around the center of mass?

The moment of inertia around the center of mass is a measure of an object's resistance to rotational motion. It is calculated by summing the products of each point mass in the object and its distance from the center of mass squared.

## 2. How do I calculate the moment of inertia using the summation formula?

To calculate the moment of inertia using the summation formula, you need to first identify all the point masses in the object and their distances from the center of mass. Then, square each distance and multiply it by the mass of the point mass. Finally, add all of these products together to get the total moment of inertia.

## 3. Can the moment of inertia be negative?

No, the moment of inertia cannot be negative. It is always a positive value since it represents the object's resistance to rotation.

## 4. Why is it important to find the moment of inertia around the center of mass?

Finding the moment of inertia around the center of mass is important because it simplifies the calculation and makes it easier to analyze the object's rotational motion. Additionally, the moment of inertia around the center of mass is the minimum value for the object's moment of inertia, meaning it requires the least amount of energy to rotate around this point.

## 5. How does the distribution of mass affect the moment of inertia?

The distribution of mass in an object affects the moment of inertia by changing how the mass is distributed around the axis of rotation. Objects with most of their mass concentrated closer to the axis of rotation will have a smaller moment of inertia, while objects with more mass distributed farther from the axis of rotation will have a larger moment of inertia.

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