# What is moment of inertia?

#### giant016

I understand how to solve for it and everything...but what exactly IS it? The term "moment" makes me think it has to do with time, but that doesn't make sense. Everwhere I look it just gives me formulas!

Staff Emeritus

#### tim_lou

moment of inertia is like mass. When you write the law of motion in polar coordinates (or in 3D, coordinates with respect to Euler angles.), you found out that there are a couple extra terms of r^2 and such, which result in the torque=I*angular acceleration. Or in Lagrangian formulation, angular momentum in planar motion is simply the generalized momentum with respect to the theta coordinate, and torque is the generalized force to the theta coordinate.

think about F=ma, in x,y direction. to accelerate something, you push in x or y direction. Now in polar coordinates, to accelerate the "angle", you push a force in the increasing theta direction, the acceleration in theta is exactly analogous to F=ma.

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#### NTUENG

Hi giant016,

I had the same problem as u when i started learning "Rotational motion of rigid bodies about a fixed axis". I also pondered for a long time over the 'moment of inertia' thing. I looked up my textbook and various sources to find out more about it. I did not want to start my tutorial on rotational motion without first understanding the underlying concepts of 'moment of inertia'. There are a few rotational cases when u need to use it. It can be applied to a system of particles or an individual particle or a mixture of both. An object can be undergoing both rotational and linear motion at the same time as in the case of a spherical object rolling down a linear slope. The moment of inertia varies according to where the axis of rotation is defined. Also there are cases when u may need to use the parallel and perpendicular axis theorems to evaluate the moment of inertia when it is at a distance away from the centre of mass of the rotating object. The shape of the object also affects the algebraic expression for the moment of inertia. For a sphere, it is 2/5 M x Radius² whereas for a rectangular plate, it is 1/12 M(length² + width²).

Hope that helps.

#### arunbg

MOI gives a measure of how mass is distributed about a particular axis of rotation. That is, for two bodies having the same mass, MOI about a particular axis will be greater for the body having its mass concentrated at greater distances from the axis.
For eg, MOI of a circular ring and disc having the same mass are respectively
MR2 and MR2/2. Can you see why the difference in magnitude arises ? You can try the same for solid and hollow spheres of same mass.

#### turdferguson

Everyone else summed it up well, MOI is the rotational equivalent of mass and depends on more than just mass. Note the R2. Mass is important, but distance from the axis is twice as important. It would be better to have twice the mass half the distance from the axis.

#### AlephZero

Homework Helper
I understand how to solve for it and everything...but what exactly IS it? The term "moment" makes me think it has to do with time, but that doesn't make sense. Everwhere I look it just gives me formulas!
"Moment" meaning in "an instant in time" comes from Latin "momentum".

But "momentum" and "moment" in the mechanics sense come from Latin "movimentum", from the verb "movere" meaning "to move".

#### kant

I understand how to solve for it and everything...but what exactly IS it? The term "moment" makes me think it has to do with time, but that doesn't make sense. Everwhere I look it just gives me formulas!

From newton , We know that the more massive the object is, the more inertia it has. In the same way. The moment is a measure of inertia for object in rotational motion. A skater would move faster as she pull her hands to her body, becases she is decreasing her moment of inertia. The moment depends on the distence from the axis of rotation, and the configuration of matter relative to this axis. The more massive the object is, the more moment it has. The greater the distence from the axis of rotation, the more moment.

#### ShawnD

I understand how to solve for it and everything...but what exactly IS it?
Basically the term "moment of inertia" describes 2 things:
1) Resistance to bending (in engineering)
2) Rotational inertia

The "moment" refers to how it's rotational. You've probably already taken statics where torque was refered to as the "moment". I think engineers say moment just so nobody else can understand what the hell they're talking about. The other 99% of the population says torque.

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