• jmf322
In summary, the conversation discusses the possibility of calculating the moments of inertia for each of the 5 spherical shells by subtracting the moment of inertia of the inner sphere from the entire sphere, and then adding them together to get the total moment of inertia. The respondent confirms that this is possible, as it is essentially what is done through integration when finding the moment of inertia of a rigid body.

#### jmf322

Hi i was wondering, if the sphere is divided into 5 spherical shells, each shell has a different density and each shell is of equal thickness

My question is: Can I calculate the moments of inertia for each shell by subtracting the moment of inertia of the sphere inside it, from the entire sphere. Does this calculate the moment of inertia for that individual shell and then I can add them up to get the entire sphere's moment of inertia? I hope I've been clear. Just not sure about adding/subtracting moments... thanks!

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Yes you can. When you're doing integration to find the moment of inertia of a rigid body that's exactly what you're doing: adding the moments of all its consitutent parts.

Yes, you can calculate the moments of inertia for each shell by subtracting the moment of inertia of the sphere inside it from the entire sphere. This is because the moment of inertia is additive, meaning that the moment of inertia of a composite object is equal to the sum of the moments of inertia of its individual parts. In this case, each shell can be considered as a separate part with its own moment of inertia, and by subtracting the moment of inertia of the sphere inside it, you are essentially isolating the moment of inertia of that specific shell. Then, by adding up the moments of inertia of all the shells, you will get the total moment of inertia of the entire sphere. I hope this helps clarify your understanding of adding and subtracting moments of inertia.

## What is the moment of inertia?

The moment of inertia is a measure of an object's resistance to changes in rotational motion. It is similar to mass in linear motion, but instead describes an object's resistance to rotational acceleration. It depends on the object's mass distribution and how it is rotating.

## How do you calculate the moment of inertia for a simple object?

For a simple object with a uniform mass distribution, the moment of inertia can be calculated using the formula I = mr², where m is the mass of the object and r is the distance from the axis of rotation to the object's center of mass.

## What is the difference between adding and subtracting moments of inertia?

When adding moments of inertia, you are combining the moments of inertia for separate objects that are rotating about the same axis. This results in a larger moment of inertia and a slower rotational speed. When subtracting moments of inertia, you are finding the difference between the moments of inertia for two objects that are rotating about the same axis. This results in a smaller moment of inertia and a faster rotational speed.

## How does the moment of inertia affect an object's rotational motion?

The moment of inertia plays a crucial role in determining an object's rotational motion. It affects how quickly or slowly an object will rotate, as well as how much torque is required to change its rotational speed. Objects with larger moments of inertia will be more resistant to changes in rotational motion.

## What are some real-world applications of adding and subtracting moments of inertia?

One common application of adding and subtracting moments of inertia is in engineering and design. By understanding how different objects' moments of inertia combine or cancel out, engineers can design more efficient and stable structures and machines. This concept is also important in sports, such as figure skating, where adding or subtracting moments of inertia can affect the athlete's rotational speed and stability.