Inertia Products: Solve Ixy, Iyz, Izx

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In summary, the conversation discusses how to evaluate triple integrals for finding moments of inertia. It also clarifies the ranges of integration for each variable.
  • #1
unscientific
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Homework Statement


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The Attempt at a Solution



I'm not sure how to find the rest; Ixy, Iyz and Izx...
Usually for integrals such as moments of inertia you will be able to reduce it to only one variable. However, there are 2 variables here; xy, yz and zx. How do i reduce it to only one?
 
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  • #2
You don't. You evaluate the triple integrals.
 
  • #3
vela said:
You don't. You evaluate the triple integrals.
Are my Ixx, Iyy and Izz are correct?

Oh, so it's dM = dx dy dz,

then the range for dx is from -√(a2-y2-z2) to √(a2-y2-z2)

for dy it's from -√(a2-z2) to √(a2-z2)

for dz it's from 0 to a

Are my ranges of integration right?
 
  • #4
unscientific said:
Are my Ixx, Iyy and Izz are correct?
No, they're not. You can't say ##y^2+z^2=a^2-x^2## because that's true only on the spherical part of the surface. Inside the hemisphere, it doesn't hold.

Oh, so it's dM = dx dy dz,
That's dV, not dM. You should write dM = k dV = k dx dy dz.

then the range for dx is from -√(a2-y2-z2) to √(a2-y2-z2)

for dy it's from -√(a2-z2) to √(a2-z2)

for dz it's from 0 to a

Are my ranges of integration right?
Yes.
 

FAQ: Inertia Products: Solve Ixy, Iyz, Izx

1. What are inertia products?

Inertia products are the mathematical representation of the distribution of mass and how it affects an object's resistance to changes in rotation. They are used to calculate the moment of inertia for three-dimensional objects.

2. How do I solve for Ixy, Iyz, and Izx?

To solve for these inertia products, you will need to use the equations for each component: Ixy = ∑mirixiyi, Iyz = ∑miriyizi, and Izx = ∑mirizixi. Here, m is the mass of each individual particle, ri is the distance from the axis of rotation to the particle, and xi, yi, and zi are the coordinates of the particle.

3. Why is it important to solve for inertia products?

Calculating the inertia products is important because it allows us to accurately predict the rotational behavior of objects. This is especially crucial in engineering and physics applications, such as designing machines and analyzing the stability of structures.

4. What factors can affect the inertia products of an object?

The inertia products of an object are affected by its mass distribution, shape, and orientation. Objects with more mass concentrated towards the axis of rotation will have smaller inertia products, while those with mass distributed farther from the axis will have larger inertia products. Additionally, the shape and orientation of an object can also impact its inertia products.

5. How can I use the inertia products to solve real-world problems?

The inertia products can be used to solve various real-world problems, such as predicting the stability and motion of rotating objects, designing machines and vehicles, and analyzing the performance of sports equipment. They can also be used in the development of new technologies and materials by helping engineers understand the effects of mass distribution on an object's behavior.

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