# Mass moment of inertia of a composite shape

In summary, the MMoI of the rectangle is 0.212 meters, but the MMoI of the quartercircle is 0.313 meters. The problem is that the parallel axis theorem must be applied twice to get the MMoI of the quartercircle about the x-axis.
Homework Statement
What is the mass moment of inertia (kg-m^2) of the following steel plate about the x-axis
Relevant Equations
Mass moment of inertia of a quarter circular plate about its base is (1/4)mr^2
Mass moment of inertia of a rectangle along its base about the x-axis is(1/3)mh^2
thickness of the steel plate is 5mm
density of steel is 7850 kg/m^3

My thought process was to get the mass moment of inertia of the rectangle and then subtract the mass moment of inertia of the quartercircle from it.
The MMoI of the rectangle is:
(1/3)(0.005*7850*.6*.3)(.3^2)= 0.212 meters
The MMoI of the quartercircle is:
(1/4)(0.005*7850*¼π 0.3^2)(.3^2) + ?
My problem is that I'm not sure how to apply the parallel axis theorem to get the Mass moment of inertia of the quarter circle about the x-axis.

The MMoI of the rectangle is:
(1/3)(0.005*7850*.6*.3)(.3^2)= 0.212 meters

My problem is that I'm not sure how to apply the parallel axis theorem to get the Mass moment of inertia of the quarter circle about the x-axis.
Pay attention to units.

To use the parallel axis theorem you need to know where the mass centre is.

I understand now, since I only know the mass moment of inertia about the base of the quartercircle, I first have to use this to find its mass moment of inertia about its centroid. Then I use this mass moment about the centroid to find the mass moment of inertia about the x-axis.
I have to apply the parallel axis theorem twice.

I understand now, since I only know the mass moment of inertia about the base of the quartercircle, I first have to use this to find its mass moment of inertia about its centroid. Then I use this mass moment about the centroid to find the mass moment of inertia about the x-axis.
I have to apply the parallel axis theorem twice.
Yes.
I looked for a way to avoid this in the present case by using some symmetry arguments, but failed.

Lnewqban said:
Take a look at problem #2 of this site:
https://owlcation.com/stem/How-to-Solve-Centroids-of-Compound-Shapes

I would leave the inclusion of the thickness and density of the material for after finding the combined centroid of the irregular flat shape.
I would not recommend to find the centroid of the entire lamina. For moments of inertia of compound shapes it is almost always simpler to find the MoI of each simple piece separately and sum. So here I would find the centroid of a semicircle, hence find the MoI of a semicircle about a tangent, etc.

## What is the definition of mass moment of inertia?

The mass moment of inertia, also known as rotational inertia, is a measure of an object's resistance to changes in its rotational motion. It is the sum of the products of each element's mass and its squared distance from the axis of rotation.

## How is the mass moment of inertia calculated for a composite shape?

The mass moment of inertia for a composite shape can be calculated by breaking the shape down into smaller, simpler shapes and using the parallel axis theorem to find the individual moments of inertia. The moments of inertia for each shape can then be added together to find the total mass moment of inertia for the composite shape.

## What factors affect the mass moment of inertia of a composite shape?

The mass moment of inertia of a composite shape is affected by the shape's geometry, the distribution of mass within the shape, and the location of the axis of rotation. The further the mass is from the axis of rotation, the higher the moment of inertia will be.

## Why is the mass moment of inertia important in engineering and physics?

The mass moment of inertia is an important concept in engineering and physics because it helps determine how much torque is needed to rotate an object, how quickly an object will rotate, and how much energy is needed to change the object's rotational motion. It is also used in the design of machines and structures to ensure stability and proper functioning.

## How does the mass moment of inertia differ from the moment of inertia for linear motion?

The mass moment of inertia is specifically for rotational motion, while the moment of inertia for linear motion is for objects moving in a straight line. The mass moment of inertia takes into account the distribution of mass in an object, while the moment of inertia for linear motion does not.

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