How Do You Calculate the Moment of Inertia for Displaced Sheets?

[alaam2005]

Homework Statement

Four sheets are displaced by d from the center and they are equal and have w by h dimensions with a mass m. It is required to find the moment of inertia around the axes x and z as shown in the figure attached.

Homework Equations

I=(mh^2/3)+(mw^2/12)

The Attempt at a Solution

http://img689.imageshack.us/img689/852/sheet.jpg

 

[ideasrule]

Moments of inertia are additive, so can you calculate the moments of inertia of the individual rectangular sheets? (Hint: use the formula at the bottom of this page: http://farside.ph.utexas.edu/teaching/301/lectures/node103.html)

 

[kerimek]

To find the moment of inertia around the x-axis, we can use the parallel axis theorem, which states that the moment of inertia of an object about an axis parallel to its center of mass is equal to the moment of inertia about the center of mass plus the product of the mass and the square of the distance between the two axes.

In this case, the moment of inertia around the x-axis would be equal to the moment of inertia of one sheet (calculated using the given dimensions and mass) plus the product of the total mass (4m) and the square of the distance between the center of mass and the x-axis (d^2). This can be represented as:

Ix = (mh^2/3)+(mw^2/12) + 4md^2

Similarly, to find the moment of inertia around the z-axis, we can use the same formula but with the distance between the center of mass and the z-axis (d) instead:

Iz = (mh^2/3)+(mw^2/12) + 4md^2

Therefore, the moment of inertia around both the x and z axes can be calculated using the given dimensions and mass, as well as the distance between the center of mass and each respective axis.