Moment of Inertia of a Dumbbell

Question

hope you are all doing well.
I am taking a course at Coursera about Robotics and in one of the quizes, I had some problem with a particular question. I passed the quiz (4/5 correct answers), however I would like to know what was wrong with my approach. The discussion forums of the course are pretty useless. I had tried them for some questions in the past and I did not get even a single response from anybody. This is the reason I am posting the question here. Here is the question:

What I did was; I first divided the dumbbell into three parts: two spheres and one cylinder. Then I calculated the mass of the cyliner (1.4074) and then the mass of one of the spheres (23.52). Then I calculated the moment of inertia for each axis of the cylinder (Ix = 0.000281, Iy = 0.0048, Iz = 0.0048). After that, I did the same thing for one of the spheres (Ix = 0.0941, Iy = 0.0941, Iz = 0.0941) and multiplied each axis by 2. Then I added this result to first cylinder and I got the matrix;
[0.1885 0 0; 0 0.1930 0; 0 0 0.1930]. However, my result was incorrect. Can you tell me what was wrong with my approach?

Chuck · Accepted Answer

First, I don't get the same mass for the spheres. Volume of a sphere is $frac{4}{3}pi r^3$, right? When I do $5600left(frac{4}{3}right)left(piright)left(0.1^3right)$ I get a different answer.
Second, if you're going to offset the centers of mass then you need to use the parallel axis theorem:
$$
I = I_{cm} + md^2 \
$$
That is, the shifted moment of inertia is the base moment of inertia (about the center of mass) and then you offset by mass times the offset distance squared.
You submitted the answer as though all objects were centered on the same spot, but they're not.
:edit:
I edited the calculation to use the radius instead of what I had originally put (the diameter). Used the right value in the actual calculation I did, but wrote the wrong numbers down here (like I've done on SO many tests)

50k4 · Answer

You have correctly identified, that you have to segment the body into 3 geometrical shapes, and calculate the inertia for all of these separately. If I understand your explanation correctly, you did not compound the inertias of the segments correctly.
Here is the Wikipedia article on how to compound the inertia of N masses. It is not just a simple summation because the inertia is dependent on geometrical coordinates. Summing up two matrices disregards this geometrical dependency.

Moment of Inertia of a Dumbbell

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