# Calculating Moment of Inertia: What Does 'r' Refer To?

• Eeduh
In summary, the equation you give for calculating the moment of inertia for a point mass refers to the distance from the axis of rotation to the point on the body that you are considering. There is a problem with this equation however because for a rigid body, you theoretically have to sum up all the point moments of inertia.

#### Eeduh

Hi,

I'm trying to teach myself some physics (dynamics in this case) and there's something I don't really get. It's how to calculate the moment of inertia.
I know the standard formula is I = m*r^2 for point masses, and I = (1/3)*m*r^2 for rigid bodies with equally divided mass, which is the case I'm interested in (I'm working on some 2d game, that's why).

Now can someone tell me what the 'r' is really referring to? Some lectures speak of the radius of the body, but I think that would be silly because then it would be the same for every point of rotation.

Is it then the distance from the point of rotation to the center of mass? This seems kind of logical, because the further away the point of rotation is from the center of mass, the more torque it'll require to rotate the object. But this would also mean that when the point of rotation is the same as the center of mass (which will be the case in many situations), moment of inertia would be 0 for r = 0, which would mean the object is infinitely easy to rotate. Makes no sense either.

Then what is r referring to? I hope someone can give me the answer.

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In the equation you give, r refers to the distance from the axis of rotation.

Allright thanks, that's at least one step in the right direction. But the distance from the axis of rotation to what? CM? Because there's a problem with that which I've allready described in my first post..

The distance from the axis of rotation to the point on the body that you are considering.

If you're considering a point mass, as you say in the first part of your first post, then r will be the distance from the mass to the axis which you are rotating the mass about.

Hmm I still don't really get it.. perhaps I should read some more on the subject. Thanks anyway.

Yeah I get it now but there remains a problem. For a rigid body, you theoretically have to sum up all the point moments of inertia. But how am I going to approach this then? For a 2d rigid body in a game, this would mean dividing the mass by the amount of pixels the object is built from, calculation the point moment of inertia for each pixel and summing it up again? There must be a better and more accurate way. please help?

## 1. What is the definition of moment of inertia?

The moment of inertia is a physical property of an object that describes its resistance to rotational motion. It is calculated by summing the mass of each particle in an object multiplied by the square of its distance from the axis of rotation.

## 2. How is moment of inertia calculated?

Moment of inertia is calculated by using the formula I = Σmr², where I is the moment of inertia, m is the mass of the particle, and r is the distance from the particle to the axis of rotation. This calculation can also be done using calculus for continuous objects.

## 3. What does 'r' refer to in the moment of inertia formula?

In the moment of inertia formula, 'r' refers to the distance from the particle to the axis of rotation. This distance is squared in the formula to account for the distribution of mass in the object and its effect on rotational motion.

## 4. How does the value of 'r' affect the moment of inertia?

The value of 'r' directly affects the moment of inertia, as it is squared in the formula. This means that particles with a larger distance from the axis of rotation contribute more to the moment of inertia than particles with a smaller distance.

## 5. How is moment of inertia used in real-world applications?

Moment of inertia is used in various real-world applications, such as designing structures and machines that are resistant to rotational forces. It is also used in fields such as physics and engineering to analyze the motion of objects and determine their stability and energy requirements.