Resistance Moment: What is the Polar Variation?

[TSN79]

In norwegian we use a term in statics/mechanics which directly translated means "resistance moment". It is denoted "W" and apparently has two directions x and y, usually written as indexes. For a circle we have
$$W_x=W_y=\frac{\pi}{32}d^3$$
What I don't understand is that there is also talk about a "polar" variation of this, which for the circle is
$$W_x=W_y=\frac{\pi}{16}d^3$$
Could someone explain to me what this "polar" variation is all about?

 

[FredGarvin]

It looks like to me you are referring to the moment of inertia. More specifically, the area moment of inertia (not to be confused with mass moment of inertia).

In two planes, X and Y, the corresponding moments of inertia are:

$$I_x = \int y^2 da$$ and

$$I_y = \int x^2 da$$. Both are a measure of an object's geometry about an arbitrary set of orthogonal axis.

The polar moment of inertia is the same as the others, but is (using the same reference notation) about the Z axis. It is the sum of the other two moments of inertia:

$$I_z = I_x + I_y = \Int (x^2 + y^2) da$$

So in your case, for the disc, $$W_x + W_y = W_z = 2*W_x = 2* W_y = 2*\frac{\pi}{32} d^3 = \frac{\pi}{16}d^3$$

In mechanics, the moment of inertia is an indication of a plate or beam's resistance to deformation due to loading. In the same sense, the polar moment of inertia is an indication of an object's resistance to torsional deformation.

 

[TSN79]

Ah, excellent! Thanks FredGarvin!

 

[morry]

Resistance moment. I think I wouldve remembered what the moment of inertia was sooner if it was called this here.