# Moment of Inertia: Derivations & Work Integrals Explained

• stunner5000pt
In summary, the conversation is about a final exam on Classical Mechanics and the speaker is looking for a resource with derivations of moments of inertia for various objects. It is mentioned that the moment of inertia for a disc of mass M spinning about an axis perpendicular to its plane and horizontal to Earth's surface is \frac{1}{2} MR^2. The speaker also asks for confirmation on the formula for the moment of inertia of a mass located at a point r from the center of the disc. The conversation also includes a question about work integrals on the exam, with the formula W = \int F \cdot d being mentioned. The speaker asks for an example of a more complicated work integral.
stunner5000pt
First of all i have a final exam tomorrow on Classical Mechanics - Cna someone point out a place that has the derivations of the moments of inertia for various objects

Now if there a disc of mass M spinning about an axis taht is perpendicular to the plane of the disc, and the plane of the disc is horizontal (parallel to Earth's surface) then it's moemnt of inertia is $\frac{1}{2} MR^2$

if there was a little mass located at a point that is r, where r<R from the center of the disc then the moment of inertia is $I = \frac{1}{2} MR^2 + mr^2$
is this correct??

Also , in class my prof said that on the exam he would have a question in which we would have to calculate a work integral... what is that ??
as far as I am concerned $W = \int F \cdot d$ is there anything more to it?? Can you point out an example of something that is more complicated liek that??

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So the little mass is standing on the disk?It's okay,then...

Daniel.

Yes, your calculation for the moment of inertia of a disc is correct. As for finding derivations of moments of inertia for various objects, a good place to start would be your textbook or class notes. You can also search online for specific objects and their moments of inertia, as there are many resources available.

A work integral is a way to calculate the work done by a variable force over a certain distance. It involves integrating the force function with respect to the distance traveled. An example of a more complicated work integral could be calculating the work done by a spring with varying stiffness over a certain displacement. In this case, the force function would be a function of displacement and you would have to integrate it over the given displacement range. I suggest practicing some examples from your textbook or class notes to get a better understanding of work integrals.

## What is moment of inertia?

Moment of inertia is a physical property that quantifies an object's resistance to changes in its rotational motion. It is often referred to as rotational inertia.

## How is moment of inertia calculated?

Moment of inertia is calculated by taking the sum of the product of each individual mass element and its distance from the axis of rotation squared. This is usually represented by the equation I = ∫ r²dm.

## What is the significance of moment of inertia?

Moment of inertia is important because it helps us understand how much torque is needed to cause a change in the rotational motion of an object. It also plays a role in determining an object's stability and its angular acceleration.

## What is the difference between moment of inertia and mass?

Moment of inertia and mass are not the same thing. Mass is a measure of an object's resistance to linear motion, while moment of inertia is a measure of an object's resistance to rotational motion. They are related, but they are not interchangeable.

## How can moment of inertia be used in practical applications?

Moment of inertia is used in many practical applications, such as designing machines and structures that need to rotate or move in a circular motion. It is also used in sports equipment, such as golf clubs and tennis rackets, to improve performance. In addition, moment of inertia is used in physics and engineering calculations to analyze rotational motion problems.

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