Moment of Inertia Tensor for a Flat Rigid Body

In summary, the conversation discusses finding the principal moments of inertia of a flat rigid 45 degree right triangle with uniform mass density. The individual attempts to solve the problem by setting up the integration boundaries and determining the elements of the moment of inertia tensor. However, it is pointed out that the limits of integration are not dimensionally correct. The individual plans to proceed with calculating all nine tensor elements and transforming it to the center of mass.
  • #1
logic smogic
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0
1. Problem
I need to find the principal moments of inertia about the center of mass of a flat rigid 45 degree right triangle with uniform mass density.

2. Useful Formulae
[tex]I_{xx}=\int_{V} \rho (r^{2}-x^{2}) dV[/tex]
[tex]I_{jk}=\int_{V} \rho (r^{2} \delta_{jk} - x_{j} x_{k}) dV[/tex]

3. Attempt at a Solution
My strategy is to set my axes so that the hypotenuse of the triangle is centered on the x-axis, with the 'right-corner' on the positive y-axis. That way, I can find the elements of the moment of inertia tensor [tex]I_{jk}[/tex] about the origin, and then translate it to the CM (1/3 up the y-axis) using the parallel-axis theorem.

If the length of one side of the triangle is "a", then using the equation for an increasing/decreasing line for the integration boundaries:

[tex]y=\pm \frac{1}{2} x + \sqrt{\frac{a}{2}}[/tex]

So,

[tex]I_{xx} = \rho \int^{\sqrt{\frac{a}{2}}}_{0} \int^{- \frac{1}{2} x + \sqrt{\frac{a}{2}}}_{\frac{1}{2} x + \sqrt{\frac{a}{2}}} y^{2} dy dx = -\frac{25}{192} a^{2} [/tex]

That should be the "x,x" element in the Moment of Inertia tensor, right?

Basically, am I setting up my integrals correctly?

(If so, I can proceed with calculating all nine tensor elements, and then diaganolizing the tensor, and transforming it to the center of mass, right?)
 
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  • #2
logic smogic said:
Basically, am I setting up my integrals correctly?
You are not. Your limits of integration are not dimensionally correct.
 

1. What is the moment of inertia tensor for a flat rigid body?

The moment of inertia tensor for a flat rigid body is a mathematical representation of the distribution of mass and shape of the object. It is a 3x3 matrix that describes how the object's mass is distributed in each direction.

2. How is the moment of inertia tensor calculated for a flat rigid body?

The moment of inertia tensor can be calculated using the object's mass, dimensions, and the parallel axis theorem. It is also possible to approximate the tensor using integration techniques.

3. What is the significance of the moment of inertia tensor for a flat rigid body?

The moment of inertia tensor is important because it determines how the object will behave when subjected to rotational motion. It provides information about an object's resistance to changes in its rotational state.

4. How is the moment of inertia tensor used in engineering and physics?

The moment of inertia tensor is used in various engineering and physics applications, such as in the design of rotating machinery, analysis of rigid body dynamics, and calculation of moments and torques in mechanical systems. It is also utilized in studying the motion of celestial bodies and understanding the behavior of atoms and molecules.

5. How does the moment of inertia tensor differ from the moment of inertia of a point mass?

The moment of inertia tensor takes into account the entire mass distribution of an object, while the moment of inertia of a point mass only considers the mass at a single point. The tensor provides a more comprehensive understanding of an object's rotational behavior, while the moment of inertia of a point mass is a simplified representation that is only applicable in certain situations.

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