Point Mass Inertia Tensor

In summary, the conversation discusses finding the moment of inertia tensor for a system of 4 particles in Cartesian coordinates and rotating the coordinates 30 degrees around the z axis. The homework equations for calculating the tensor components are also mentioned. It is suggested to try calculating the components in the rotated system to get the correct solution.
  • #1
darthmonkey
6
0

Homework Statement



A)Find the moment of inertia tensor for a system of 4 particles of mass 1kg at A=(1,1,0), B=(1,-1,0) C=(-1,1,0) D=(-1,-1,0) in Cartesian coordinates.

B)Rotate the coordinates 30 degrees around the z axis and find the tensor in the new coordinates.

Homework Equations



Ixx=[itex]\sum[/itex]y^2+z^2
Iyy=[itex]\sum[/itex]x^2+z^2
Izz=[itex]\sum[/itex]y^2+x^2
Ixy=[itex]\sum[/itex]xy
Iyx=[itex]\sum[/itex]yx
Izx=[itex]\sum[/itex]zx
Izy=[itex]\sum[/itex]zy

I'=RIR*

The Attempt at a Solution


I get Ixx=4, Iyy=4, and Izz=8. Anything multiplying z is 0 and all the xy components cancel out. So a diagonal matrix.

For B I just apply the z rotation matrix with 30,RIR*, but when I do this I get the same thing. This can't be right so I either calculated my starting matrix wrong or am rotating incorrectly.
 
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  • #2
Hi darthmonkey. Maybe the tensor components don't change under the rotation! Have you tried figuring out the new coordinates of the particles in the rotated system and then calculating the components of the tensor from these components?
 
  • #3
Haha, thanks. I got it to all work out.
 

What is a point mass inertia tensor?

A point mass inertia tensor is a mathematical representation of the mass distribution and rotational characteristics of a point mass in three-dimensional space.

Why is the point mass inertia tensor important?

The point mass inertia tensor is important because it allows scientists and engineers to calculate the rotational dynamics of a point mass and understand how it will behave when subjected to external forces.

How is the point mass inertia tensor calculated?

The point mass inertia tensor is calculated by multiplying the mass of the point mass by a matrix of inertia values, which depend on the distance of the point mass from each axis of rotation.

What is the relationship between the point mass inertia tensor and the moment of inertia?

The point mass inertia tensor is a matrix representation of the moment of inertia, which is a property of an object that determines its resistance to rotational motion.

How is the point mass inertia tensor used in real-world applications?

The point mass inertia tensor is used in various fields such as physics, engineering, and robotics to calculate the rotational dynamics of objects and design systems that can control or manipulate rotational motion.

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