Fundamentals of the Moment of Inertia

http://en.wikipedia.org/wiki/Moment_of_inertia

Yep, start with Wikipedia and then come on back with more specific questions. This is a pretty broad topic.

Like many, as a young injuneer I had difficulty quantifying inertia in my mind. For instance what might the inertia of a complete automobile be, say bout it's vertical axis? Then I started thinking in terms of radius of gyration and it became a lot easier. I found I could guess the inertia of a complete vehicle with a modest degree of accuracy. If you concentrated all the mass of a vehicle into two eqivalent masses either side of the centre (of gravity), how far apart would they be?? 1/4 of the length of the vehicle? 1/3rd?? Something in that region. Multiply that distance by the mass and you have a reasonable estimate.

It's not difficult to imagine the relative proportions of the inertias of a vehicle is it? It's clear that the inertia about the vertical axis and that about the lateral axis are going to be about the same right, and that about the longitudinal axis quite a bit smaller. Think about the relative radii of gyration. The mass is the same in all cases, but the distance your two masses must be apart if you look at the thing from the front is clearly a lot less than either of the other views.

In a nutshell, when something with mass moves it has inertia. Rotation about an axis is a type of movement. So in order to describe the inertia inherent in a rotating body, there is a quantity called moment of inertia. It basically depends on the rotation axis and the distribution of mass about that axis. For instance, a solid ball of mass M and radius R spinning about an axis through its center has a moment of inertia of 2/5 MR² whereas a hollow cylinder of mass M under the same circumstances has a moment of inertia of 2/3 MR². Kinetic energy can be described in terms of moment of inertia by the formula:

K.E. = ¹/₂Iω²

where ω is the angular velocity and I is the moment of inertia.

So clearly a hollow sphere of mass M has a higher moment of inertia than a solid sphere of mass M and if they have the same angular velocity, then the Hollow sphere has larger kinetic energy.

Hope that helps.

Moment of inertia has to do with the strength of an object and how its shape affects its strength properties. The great application of moment of inertia is the "I" beam. Where you can greatly reduce the mass of your beam, but still maintain strength by having much of the mass far away from the beam's centre.

Hmmm. IMO that's confusing for someone trying to grasp the concept od inertia. The second moment of area of a cross-section has nothing to do with kinetics.

a very basic experience: take a long straight stick.

1) Make it rotate around its axis (the centre line parallel to its length),

2) then make it turn around any direction perpendicular to the previuos one and feel how much more strength you have to use:

Now can you visualize better the meaning of the moment of inertia ?

the body is the same, but in different directions the moment of inertia is different.

play with the umbrella:

take it closed and make it turn around the axis,

open it and repeat the experience . . .

In structural engineering the moment of initera is used to design non-eccentric connection. In semetric shapes the Ix and Iy is the centroid of that shape. In shapes that are not semetric we use the princapls of the moment of initera to locate the centroid and apply loads in this location. Ultimately stopping eccenticity, allows for simple connections to be designed this reduces the costs of connections. You can calculate where the moment of inertia is by summing the moments so at any location the opp. moment is created in the same distance from the point

silly figure-1 Silly figure-2

semetric bolt arragment eccentric connection

x x X X

x(centroid at 1/2 width& height X (.)centroid not at 1/2 width

at 1/2 height

x x X X

inertia is something like a form of resistance. resistance to change. getting up in the morning needs some energy to overcome the inertia...
in mechanics inertia for example is involved as a resistance of a beam to deform. A beam with a high cross sectional area provides more resistance to deformation. in motion a high inertia will indicate greater resistance to change in motion. Mass is inertia in rectilinear motion, while for angular motion the mass moment of inertia indicates that resitance to change the mode of motion. a high inertia means a larger force or moment is necessary to accelerate an object. in ship stability a high area moment of inertia of the ship waterplane area about an axis indicates the ship's resistance to rotate about that axis. a huge area moment of inertia about the longitudinal axis for example indicates a high resistance against rolling. a high area moment of inertia about the transverse axis indicates a high resistance to pitching or purpoising. it is difficult indeed to visualize inertia, but it can be thought of as a form of resistance, resistance to change.

Watch a figure skater spinning. When she puts her arms out, she increases moment of inertia and spins slowly. When she pulls in her arm, the moment of inertia is smaller and she spins faster.

Great demonstration of rotational/angular momentum, next stop gyroscopics.

Next week, I'll ask her to pull in the other arm.