Physical Meaning of Product of inertia

[SpartanG345]

I was reading about products of inertia in my dynamics book, it was defined really well, however it did not explain the physical meaning of the product of inertia.

Is it still resistance to rotation? However products of inertia can be negative. Are they simply mathematical artefacts with no definite physical meaning?

 

[tiny-tim]

Hi SpartanG345!

… Are they simply mathematical artefacts with no definite physical meaning?

Basically, yes.

An individual product of inertia (I_xy etc) has no physical consequence.

(though one could probably work out a physical, or rather geometrical, meaning for the definition of it)

 

[Studiot]

Yes indeed there is a physical significance.

A simple dynamics treatment of rotational inertia sets the axis of rotation as one of the coordinate axis - x, y or z.

In this case the rotational inertia is simply $${I_{xx}}$$, $${I_{yy}}$$ or $${I_{zz}}$$.
There is no contribution from the moments about the other axes.

If we wish to generalise the rotation to one about some arbitrary axis, $$\omega$$, inclined at $$\alpha$$ to the x axis, $$\beta$$ to the y and $$\gamma$$ to the z then the rotational inertia about that axis is given by

$${I_\omega } = {I_{xx}}{\cos ^2}\alpha + {I_{yy}}{\cos ^2}\beta + {I_{zz}}{\cos ^2}\gamma + 2{I_{yz}}\cos \gamma \cos \beta + 2{I_{zx}}\cos \alpha \cos \gamma + 2{I_{xy}}\cos \alpha \cos \beta$$

Note there are contributions from

1)moments about the individual coordinate axis – these are called moments of inertia

The moment of inertia of a body about an axis is defined as the sum taken over all particles of the body
$$\sum {m{r^2}}$$
Where m is the mass of any particle and r is its distance from the axis.
(This is not the same as the second moment of area by the way.)

2)The product of inertia of a body about two planes which is defined as the sum taken over all particles of the body
$$\sum {mxy}$$
Where m is the mass of any particle and x,y is its perpendicular distance from the planes.

It is worth noting that these objects are more complicated than simple vectors. Vectors can be resolved into mutually exclusive contributions referred to each axes, called components. These objects cannot. There are cross contributions or interactions from the other two axes, when we isolate anyone of them.
The direct contributions are called the moments of inertia.
These cross contributions are called the products of inertia.

 

[tiny-tim]

… However products of inertia can be negative. Are they simply mathematical artefacts with no definite physical meaning?

2)The product of inertia of a body about two planes which is defined as the sum taken over all particles of the body
$$\sum {mxy}$$
Where m is the mass of any particle and x,y is its perpendicular distance from the planes.

It is worth noting that these objects are more complicated than simple vectors. Vectors can be resolved into mutually exclusive contributions referred to each axes, called components. These objects cannot. There are cross contributions or interactions from the other two axes, when we isolate anyone of them.
The direct contributions are called the moments of inertia.
These cross contributions are called the products of inertia.

So an individual product of inertia (eg I_xy) is simply a mathematical artefact with no physical consequence?

 

[Studiot]

Sorry Tim but it has great physical significance for balancing rotating bodies such as wheels or turbine blades

Rotation aboout any axis generates angular momentum.
This is represented by the angular momentum vector.
If the axis of rotation is aligned with one of the principal axes of the body the angular momentum vector is also aligned in the same direction and the rotation is stable.
If the rotational axis is not principal then a torque, applied at right angles to this axis, is generated and causes the direction of momentum vector to describe a cone in space as the body rotates.
If the body is a rotating wheel with bearings this torque creates a reaction against the bearings.
Any unbalance in the distribution of the mass will be lead to vibration of the wheel and perhaps eventually to destruction.

Of course the directions of the principal axes are those where the products of inertia vanish as the arbitrary rotational axis varies.

 

[tiny-tim]

Yes, SpartanG345 and I know all that …

but the point that he is making is that I_xy, say, is part of a larger whole (which does everything you've just described), but has no individual physical consequence.

It is not like the component of a vector in a particular direction … for example, mgcosθ occurs in its own right in plenty of equations, such as (block on a slope:) F - mgcosθ = ma.

Can you find an equation in which I_xy similarly features?

If not, then surely I_xy is "simply a mathematical artefact with no definite physical meaning" ?

 

[Studiot]

The products of inertia contain information about the disposition of mass in the system, (as do the moments), otherwise they could not modify the expression for the rotational inertia. There is more than one way to obtain a given set of moments of inertia, the distinguishing information is provided by the products of inertia.
Therefore they are significant.

I will try to think of a simple example, probably using a system of particles) to show what I mean.

 

[Studiot]

Consider the simple two particle systems in the sketches.

Despite the different mass distribution they have identical moments of inertia so the only way to distinguish is via the products of inertia.

I have kept to 2D for simplicity.

 

[tiny-tim]

Despite the different mass distribution they have identical moments of inertia so the only way to distinguish is via the products of inertia.

Yes, you've drawn one system with I_xx = 2mb² and I_y = 2ma² but I_xy = 0, all about axes through the centre of mass,

and another system with the same mass (2m), and with the same I_xx and I_yy but with I_xy = 2mab, all about axes not through the centre of mass.

But what physical consequence do you say should be attributed to the extra I_xy ? ​

 

[Studiot]

Exactly what I said in post#7.
That knowledge of moments of inertia alone is insufficient to fully describe the rotational dynamics of either system or indeed any system.

If you do not consider that to be a matter of physical significance I will leave it as a matter of opinion, I certainly do.

 

[tiny-tim]

That knowledge of moments of inertia alone is insufficient to fully describe the rotational dynamics of either system or indeed any system.

Well, that's obvious, and it's what SpartanG345 and I have both been saying.

But the fact that you need to know the products of inertia together with the other entries in the inertia tensor doesn't mean that a product of inertia itself has an identifiable physical consequence.

If you do not consider that to be a matter of physical significance I will leave it as a matter of opinion, I certainly do.

The entries in the moment of inertia tensor (ie the moments of inertia and the products of inertia) as a whole certainly have physical consequences.

But (unlike, say, a component of a force) a product of inertia on its own does not appear to have any physical consequence. If so, it is a "mathematical artefact with no definite physical meaning"

 

[D H]

Products of inertia certainly do have physical meaning. Suppose the I_xy for some object is non-zero. That means that the object cannot perform a pure x-axis rotation without some external torque. (The same also pertains to a pure y-axis rotation.)

 

[zlovering]

Saying that the entires in the moment of inertia tensor have no physical consequence on their own is exactly equivalent to saying that mass has no physical consequence on its own since it occupies positions in the mass tensor. After all, [Torque] = [Inertia tensor] x [Angular acceleration] is the rotational equivalent to [Force] = [Mass tensor] x [Acceleration].

 

[nvn]

Products of inertia certainly do have physical meaning.

Very well said. A nonzero product of inertia causes a body to twist about its longitudinal axis when you try to rotate the body about one of its transverse axes. This is just another way of saying what D H wrote.

 

[skeleton]

1) Consider
gam = density
V = volume
Then
m = mass = gam*v

2) Similarly,
E = elastic modulus
I = moment of inertia
Then
K = stiffness = E*I

"I" by itself is spatial but not physical, where as, "E*I" is physical. "I" is a spatial property of an object to conveys how it will react in matters of inertial forces, be they linear or rotational.

 

[zlovering]

Spatial and physical are not exclusive adjectives. This conversation has transformed into a matter of semantics. If you choose to define physical as "any aspect of an object or substance that can be measured or perceived without changing its identity" (as Wikipedia does, not that Wikipedia is right all the time, but I think this is a pretty fair definition) then clearly density, volume, second moment of area, moment of inertia, mass, modulus, and stiffness are all "physical" properties.

Furthermore, in reply skeleton, the moment of inertia you refer to is not the moment of inertia we are discussing in this thread, but rather the second moment of area with units length^4. In special circumstances, namely for homogeneous prismatic bodies, this value is related to the mass moment of inertia (units mass*length^2), but then you have to include density in the calculation, making it NOT purely "spatial". Secondly, just because two equations have the same format, that is a=b*c, does not mean that they are analogous.

 

[kalaagh]

Product of inertia has actually a physical meaning.

In general I_ij(=I_ji) is the inertia of a mass rotating around the i axis against its rotation around the j axis. For example consider a mass rotating around the x axis. Then I_xx is its inertia against rotation around the x-axis while I_xy is its inertia against rotation around the y axis.