# Moment of Inertia Problem: Perpendicular Axes THM

• ultimateceej
In summary, the conversation discusses a moment of inertia problem involving the perpendicular axes theorem. The problem asks to prove the theorem and use it to find the moment of inertia for a thin, uniform square sheet through any axis in its plane. The result is found to be 1/2 ML^2, but there is confusion over a possible misprint in the book. It is later confirmed by a physics teacher that the correct answer is indeed 1/2 ML^2.
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?

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

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.

## 1. What is the Moment of Inertia Problem: Perpendicular Axes Theorem?

The Moment of Inertia Problem: Perpendicular Axes Theorem is a principle in physics that relates to the calculation of the moment of inertia of a rigid body. It states that the moment of inertia of a body about an axis perpendicular to its plane is equal to the sum of the moments of inertia of the body about two perpendicular axes that lie in its plane.

## 2. How is the Perpendicular Axes Theorem used in real-life applications?

The Perpendicular Axes Theorem is used in a variety of real-life applications, such as in the design of structures and machines. It is also used in the study of rotational motion in physics and engineering, including the analysis of moments of inertia in rotating objects like wheels and flywheels.

## 3. What is the difference between the perpendicular axes theorem and the parallel axes theorem?

The perpendicular axes theorem and the parallel axes theorem are both principles that relate to the calculation of the moment of inertia of a rigid body. However, the perpendicular axes theorem applies when the axis of rotation is perpendicular to the plane of the object, while the parallel axes theorem applies when the axis of rotation is parallel to the plane of the object.

## 4. How is the moment of inertia of an object affected by its shape?

The moment of inertia of an object is affected by its shape because it is a measure of an object's resistance to changes in its rotational motion. Objects with larger moments of inertia require more torque to change their rotational motion, and this is influenced by the distribution of mass in the object's shape.

## 5. What are the units of moment of inertia?

The units of moment of inertia are kilogram-meter squared (kg*m2). This is because the moment of inertia is calculated by multiplying the mass of an object by the square of its distance from the axis of rotation, which has units of kilograms and meters squared.

• Introductory Physics Homework Help
Replies
1
Views
873
• Introductory Physics Homework Help
Replies
11
Views
1K
• Introductory Physics Homework Help
Replies
25
Views
426
• Introductory Physics Homework Help
Replies
21
Views
1K
• Introductory Physics Homework Help
Replies
2
Views
597
• Introductory Physics Homework Help
Replies
28
Views
526
• Introductory Physics Homework Help
Replies
3
Views
842
• Introductory Physics Homework Help
Replies
15
Views
631
• Introductory Physics Homework Help
Replies
52
Views
2K
• Introductory Physics Homework Help
Replies
13
Views
1K