Question 3: Symmetry in the moment of inertia tensor (25 points) The figure abov

Question

Question 3: Symmetry in the moment of inertia tensor (25 points) The figure above shows 4 identical cylinders meeting in a cross formation. (Saying that they are identical means that they have the same mass, same radius, and same length. Just in case you wanted to ask the exam proctor if identical means, like, the same.) They meet at 90 degree angles. Additionally, two of them have little spheres stuck to the ends; assume that the spheres are identical to each other, and each is attached to the center of one of the ends. I am not telling you any of the masses, lengths, or radii. You don't need them for what I'm asking (a) (10 points) Which elements of the moment of inertia tensor are zero? Which (if any) non-zero elements are equal to each other? Why? Note that I'm not asking you to say Such-and-such elemen is 0.752 or whatever. I'm just asking you to say These elements are zero because [reason]" and These two for three, or whatever) are equal to each other because [reason].”

Explanation / Answer

(a) just by looking at the object we can say that Iyz and Izy component of the moment of inertia tensor will be zero. The reason being the distance from y-axis is zero for the cylinder with the sphere's at the end, while the distance from the Z-axis is zero for cylinder without the spheres. Iyz=Izy = integral(YZdm)

(b) The torque is zero which means the angular momentum is conserved. But the angular velocity will not be constant as that happens only when the angular velocity is along the axis of maximum or minimum moment of inertia.

Angular momentum is conserved. Angular velocity may not be conserved. A solid body has three moments of inertia, which are orthogonal to each other. If the body is rotating about its axis of maximum moment of inertia or its axis of minimum moment of inertia, then the rotation is stable and the angular velocity will be constant. Otherwise, the rotation will not be stable, even though the angular momentum must be conserved.

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