# Simple Moment of Inertia Question

• driven4rhythm
In summary, the problem is that the equation for moment of inertia is not in the book that the student is using.
driven4rhythm
Here is the picture of the problem: http://yfrog.com/3uwtfup
I understand every other step except for the equation for moment of inertia. In my book the most basic equation for moment is ΔI = (r)^2 Δm but for this problem ΔI = (1/2)(r)^2 Δm. Why is that? Does it have something to do with the method of integration?

When you are integrating to calculate the moment of inertia (MOI), you take the MOI of a tiny mass and sum all of those to get the total. MOI of a point mass is mr^2. In this case the tiny mass is a disk. MOI of a solid cylinder, disk etc is 1/2 mr^2.

When you are integrating to calculate the moment of inertia (MOI), you take the MOI of a tiny mass and sum all of those to get the total. MOI of a point mass is mr^2. In this case the tiny mass is a disk. MOI of a solid cylinder, disk etc is 1/2 mr^2.
How would you go about deriving the constant that's out front, in this case 1/2?

Using integration :)
There is an example or two in almost every physics text that i have seen. Check out some other books if your book doesn't have it.

The constant comes from the fact that a solid cone is made up of many thin disks (horizontal cross-sections of the cone wrt axis of rotation). So the rotational inertia of the solid cone is equal to the total rotational inertia of all these thin disks. The rotational inertia of a disk of radius R about its center of mass is
$$I_{disk}$$ = 1/2 M$$R^{2}$$

Which is derived from the rotational inertia of a ring of radius R about its center of mass
$$I_{ring}$$ = M$$R^{2}$$

Which is derived from the rotational inertia of a point mass at a radius R from the axis of rotation.
$$I$$ = M$$R^{2}$$

So a small part of the rotational inertia of the cone (d$$I_{cone}$$) is equal to the rotational inertia of a thin disk of radius y, 1/2 dm$$y^{2}$$.

## 1. What is moment of inertia?

Moment of inertia is a measure of an object's resistance to changes in its rotational motion. It is determined by the object's mass, shape, and distribution of mass.

## 2. How is moment of inertia calculated?

The moment of inertia for a simple object can be calculated by using the formula: I = mr², where I is the moment of inertia, m is the mass of the object, and r is the distance from the axis of rotation to the object's mass.

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

Moment of inertia measures an object's resistance to rotational motion, while mass measures an object's resistance to linear motion. In other words, moment of inertia takes into account the distribution of mass in an object, while mass only considers the total amount of mass.

## 4. How does moment of inertia affect an object's motion?

The moment of inertia directly affects an object's rotational motion. Objects with a higher moment of inertia will be more difficult to rotate, while objects with a lower moment of inertia will be easier to rotate.

## 5. What is an example of a simple moment of inertia question?

An example of a simple moment of inertia question would be: "A rod with a mass of 2 kg and length of 5 m is rotating about its center with an angular velocity of 4 rad/s. What is the moment of inertia of the rod?"

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