Calculating Rotational Inertia of a Rectangular Prism

[r16]

Homework Statement

There is a rectangular prism of uniform mass distribution with lengths of $a$, $b$, and $c$ (b>a>c). Calculate it's rotational inertia about an axis through one corner and perpendicular to the large faces.

Homework Equations

$$I = \int r^2 dm$$
$$r^2 = x^2 + y^2 + z^2$$
$$\rho = \frac{M}{V}$$
$$V = abc$$

The Attempt at a Solution

I am examining a cubic differential mass of $dm$'s contribution on the moment of inertia about the axis of rotation. The radius between $dm$ and the axis of rotation is $r^2 = x^2 + y^2 + z^2$. The density, $\rho$, is constant which is $\frac{M}{V}$, so $dm = \rho dV$.

$$I = \int r^2 dm = \int (x^2 + y^2 + z^2) \rho dV$$
[tex] I = \rho \iiint_V x^2 dV + y^2 dV + z^2 dV = \int^a_0 \int^b_0 \int^c_0 x^2 dzdydx + \int^a_0 \int^b_0 \int^c_0 y^2 dzdydx + \int^a_0 \int^b_0 \int^c_0 z^2 dzdydx [/itex]
$$I = \frac{\rho}{3} ( a^3 bc + ab^3 c + abc^3)$$
$$I = \frac{M}{3abc} ( a^3 bc + ab^3 c + abc^3)$$
$$I = \frac{M}{3} (a^2 + b^2 + c^2)$$

This problem looked cool so I did it, but it was an even one so there is no answer in the back of the book. Does this look right?

 

[malawi_glenn]

Is this the situation? (red is axis of rotation)

If this is the case, the moment of inertia can not depend on c.
There is another, simpler, shape that you can use instead...

 

[Delta2]

The classical mistake when calculating MoIs: $r$ should not be the magnitude of the position vector of the element dV, but its distance perpendicularly to the axis of rotation.

 

[Orodruin]

Is this the situation? (red is axis of rotation)
View attachment 303793
If this is the case, the moment of inertia can not depend on c.
There is another, simpler, shape that you can use instead...

Just to add a side note: This depends on what is considered given. If the density is known and fixed, then the MoI will depend on c because larger c means more mass. If the density is instead adjusted such that the total mass $M$ is known, then indeed the MoI will be independent of c.

 

[malawi_glenn]

This depends on what is considered given

That's true, the c-dependece will show up in M if a fixed density is given (also M would have a and b dependence M = ρabc)