How Is Moment of Inertia Calculated for Arbitrary Axes in Dynamics?

[Rhi]

Homework Statement

Hi, I've got a Dynamics question on moments of inertia and I'm not really sure what the question is trying to get at. The question is

let R=(O,B) be a rest frame of a rigid body and let the entries in J(R) be Jij. Show that the moment if inertia about an axis through O in the direction of the unit vector with components xi is Jijxixj.

Homework Equations

The definition that we have been given is that the diagonal entries of the matrix J(R) A,B and C are the moments of inertia about the x, y, z axes respectively.

The Attempt at a Solution

I know that if you have two rest frames R(O,B) and R'(O,B') and H transition matrix from B' to B then Jij=HipJpq'Hjq. Am I supposed to use this? I'm really confused..

 

[anathema]

Dear student,

Thank you for reaching out for help with your Dynamics question. I understand that you are unsure about what the question is asking and how to approach it. Let me try to break it down for you.

First, let's define some terms to make sure we are on the same page. A rest frame is a reference frame in which the rigid body is not moving. J(R) is the inertia tensor, which is a mathematical representation of the distribution of mass in a rigid body. It is a 3x3 matrix with entries Jij, where i and j represent the x, y, and z axes. The diagonal entries of J(R) are known as the moments of inertia, which represent the resistance of a body to rotational motion about a particular axis.

Now, let's look at the actual question. The question is asking you to show that the moment of inertia about an axis through O (the origin of the rest frame) in the direction of a unit vector with components xi is equal to Jijxixj. In other words, it is asking you to prove the formula for calculating the moment of inertia about an arbitrary axis.

To do this, you can use the definition of the inertia tensor that you were given, which states that the diagonal entries of the matrix J(R) are the moments of inertia about the x, y, and z axes respectively. You can also use the formula that you mentioned, Jij=HipJpq'Hjq, where H is the transition matrix from one rest frame to another.

I hope this helps clarify the question for you. If you need further assistance, please don't hesitate to ask. Good luck with your studies!