Moment of inertia tensor of three spheres

[Silken]

Hi everyone

Homework Statement

I want to find out the moment of intertia tensor of the graphic below.

Homework Equations

parallel axis theorem

The Attempt at a Solution

We know the moment of inertia for one sphere, that's given, so I don't have to calculate it explicit.Now I have trouble understanding the moment of inertia tensor. It looks like the following:

$$I=\begin{pmatrix} I_{xx} & I_{xy} & I_{xz} \\ I_{yx} & I_{yy} & I_{yz} \\ I_{zx} & I_{zy} & I_{zz} \end{pmatrix}$$

I understand that Ixx, Iyy and Izz are the moments of inertia regarding to x y and z but I don't understand what Ixy Ixz Iyz etc. 'means'. I just can't picture it. Furthermore I tried to find the xx yy and zz, I found out :

$$I_{xx}=I_{yy}=\frac {54} {5} MR^{2}+9r^{2}m$$

and

$$I_{zz}=\frac 4 5 MR^{2}+(R+r)^{2}M$$

But how do I find the xy xz etc. I know I just have to calculate thre more, because the tensor is symmetrical. But I don't know how do to it. Is my solution right thus far?

Thanks for your help

 

[vela]

Hi everyone

Homework Statement

I want to find out the moment of in[STRIKE]t[/STRIKE]ertia tensor of the graphic below.

Homework Equations

parallel axis theorem

The Attempt at a Solution

We know the moment of inertia for one sphere, that's given, so I don't have to calculate it explicitly. Now I have trouble understanding the moment of inertia tensor. It looks like the following:

$$I=\begin{pmatrix} I_{xx} & I_{xy} & I_{xz} \\ I_{yx} & I_{yy} & I_{yz} \\ I_{zx} & I_{zy} & I_{zz} \end{pmatrix}$$

I understand that Ixx, Iyy, and Izz are the moments of inertia regarding to x, y, and z, but I don't understand what Ixy, Ixz, Iyz, etc. [STRIKE]'[/STRIKE]mean[STRIKE]s'[/STRIKE]. I just can't picture it. Furthermore I tried to find the xx, yy, and zz, and I found out :

$$I_{xx}=I_{yy}=\frac {54} {5} MR^{2}+9r^{2}m$$

I take it M is the mass of the large spheres and m is the mass of the small sphere. How did you get 9mr²? Surely, the contribution of the small sphere must also depend on R. Also, you shouldn't have I_xx=I_yy as far as I can see.

and

$$I_{zz}=\frac 4 5 MR^{2}+(R+r)^{2}M$$

What happened to the small sphere? There's no m in your result.

But how do I find the xy, xz, etc. I know I just have to calculate three more, because the tensor is symmetrical. But I don't know how do to it. Is my solution right thus far?

Thanks for your help

Crank out the integrals like
$$I_{xy} = \int xy\,dm$$