Moment of Inertia Problem: Perpendicular Axes THM

[ultimateceej]

[SOLVED] Moment of Inertia Problem: Perpendicular Axes THM

Homework Statement

First, the problem asked us to basically prove the Perp. Axes Thm and then use it to prove that the MoI of a thin, uniform square sheet with mass M and side L through ANY axis in its plane is equal to 1/2 M L^2

Homework Equations

The moment of inertia for a square sheet of a perpendicular axis through its center of mass is 1/6 M L^2

The Attempt at a Solution

First, I tried to simply say that I_x is perp. to I_y and that their sum is I_o which is equal to 1/6 M L^2 (above). And since the square is uniform, I_x = I_y = 1/2 I_o but this gave me I_x = 1/12 M L^2. Is it possible that the book has a misprint? How is it possible for the moments of inertia of an axis in the plane to be 3 times the MoI through the center of mass?

 

[ultimateceej]

Nevermind, physics teacher confirms the the typo. The correct answer is what I got, I_x = 1/12 M L^2

 

[Moetasim]

The Perpendicular Axes Theorem states that the moment of inertia for a planar object through any axis perpendicular to the plane is equal to the sum of the moments of inertia about two perpendicular axes lying in the plane and intersecting at the given axis. This theorem is useful in simplifying the calculation of moments of inertia for complex shapes.

In the given problem, we are asked to prove that the moment of inertia for a thin, uniform square sheet through any axis in its plane is equal to 1/2 ML^2. To solve this problem, we can use the Perpendicular Axes Theorem. First, we can calculate the moment of inertia through the center of mass of the square sheet, which is given by 1/6 ML^2. Then, we can use the theorem to find the moments of inertia about two perpendicular axes in the plane, which intersect at the given axis. Since the square sheet is uniform, these two moments of inertia will be equal and can be denoted as I_x and I_y. Using the Perpendicular Axes Theorem, we get I_x + I_y = 1/6 ML^2. Since I_x = I_y, we can rewrite this as 2I_x = 1/6 ML^2, or I_x = 1/12 ML^2. This is the moment of inertia about one of the perpendicular axes in the plane. However, since the square sheet is symmetric, the moment of inertia about the other perpendicular axis will also be equal to 1/12 ML^2. Therefore, the total moment of inertia about the given axis is equal to 1/12 ML^2 + 1/12 ML^2 = 1/6 ML^2, which is equal to 1/2 ML^2 as desired.

In conclusion, it is not a misprint in the book. The moments of inertia about the two perpendicular axes in the plane are not 3 times the moment of inertia through the center of mass, but rather equal to 1/12 ML^2 each. The Perpendicular Axes Theorem allows us to find the moment of inertia about any axis in the plane by using the moments of inertia about two perpendicular axes.