Moment of inertia of a thin, square plate

[simphys]

Homework Statement

the moment of inertia about an axis through the center of and perpendicular to a uniform, thin square plate. mass M and dimension L x L.

Relevant Equations

d

I don't really understand what the 2 integrals (dx and dxdy) for I_x represent. Could I get some explanation here please? Thanks in advance.

 

[kuruman]

The second line in the derivation of $I_x$ has one $dx$ too many. Is that what bothers you?

 

[TSny]

Also, the second line is missing the $r^2$ and the limits of integration for $x$ are not correct. Keep in mind that this integral represents the moment of inertia about the x-axis, and the x-axis lies in the plane of the thin plate. Think about how to express $r^2$ in terms of $x$ and/or $y$.

 

[hutchphd]

If you put in the correct $$r^2=x^2+ y^2$$ in the second line then you see that the x and y integrals give you the "perpendicular axis theorem" without issue. Please redo this.

 

[simphys]

Also, the second line is missing the $r^2$ and the limits of integration for $x$ are not correct. Keep in mind that this integral represents the moment of inertia about the x-axis, and the x-axis lies in the plane of the thin plate. Think about how to express $r^2$ in terms of $x$ and/or $y$.

Yep, that is what I did I expressed in terms of y^2 and then went ahead with that. But I just didn't understand how this solution came about basically.

 

[simphys]

If you put in the correct $$r^2=x^2+ y^2$$ in the second line then you see that the x and y integrals give you the "perpendicular axis theorem" without issue. Please redo this.

this is the solution. not mine :)

 

[hutchphd]

So where is your attempt? How are we to help?

 

[simphys]

So where is your attempt? How are we to help?

well.. that is exactly what I don't understand..
I didn't understand why the solution has used two integrals in such a way.. I haven't used a volume integral or smtn. I did differently by using a mass element and summing over that with one definite integral.

 

[hutchphd]

One must sum each mass element (mass density times volume element) over the entire volume of the object and scaled by the square of the distance to the chosen axis . The world is three dimensional and so is the integral. Show us your work.