Rigid Body: Rotation of a coordinate axes made up of rods

[^_^physicist]

Homework Statement

A rigid body consists of three uniform rods each of mass, and length 2a, held mutually perpendicular at their midpoints (choose and axis along the rods).

a) Find the angular momentum and kinetic energy of the body if it rotates with angular velocity w about an axis passing through the origin, and for passing though a point (1,1,1).

b) Show that the moment of inertia of inertia is the same for any axis passing through the center

c) prove the results of example 9.11 (ignore this one...I think I can do it without a problem).

Homework Equations

Products of inertia (I_xx, I_yy, I_zz, I_xy, I_xz, I_yz). to determine moment of intertia tensor

L= w*I(tensor).

Kinetic Energy= all rotational kinetic energy so: T_rot = 1/2* (I_xx*(w_x)^2 + I_yy*(w_y)^2 + I_zz*(w_z)^2 + 2*I_xy*w_x*w_y+2*I_xz*w_x*w_z +2*I_yz*w_z*w_y).

The Attempt at a Solution

If I could figure out the inertia tensor, I am set on my way; however, I am hung up. I have thought about using the parrell and perpendicular axis theorems to take advantage of the fact that all portions of the form are constucted from the three perpendicular rods; however, I am not sure how to do this.

I did, however, make an attempt using the parallel axis theorem:

I_cm=ma^2=I_z. I_cm= 1/12ML^2 which implies=> I_3= (4*a^2*m)/3, and similarly for I_1. Noting that I_2 is the axis of rotation, for the first portion, I_2=0. I_total=(8a^2*m)/3.

I would much rather use the products of inertia instead however.

 

[Jhero]

Hello, thank you for your post. Here are some suggestions for finding the moment of inertia tensor for this system:

1. Start by drawing a diagram of the system and labeling the three rods as A, B, and C.

2. Use the parallel axis theorem to find the moment of inertia for each rod about its own center of mass.

3. Then, use the perpendicular axis theorem to find the moment of inertia for each rod about an axis perpendicular to its length.

4. Finally, use the parallel axis theorem again to find the moment of inertia for each rod about the axis passing through the center of the system.

5. Add all of these individual moments of inertia together to find the total moment of inertia tensor for the system.

As for the other parts of the problem, for part a) you can use the formula for angular momentum and kinetic energy provided in the post, and plug in the values you found for the moment of inertia tensor.

For part b), you can show that the moment of inertia is the same for any axis passing through the center by using the fact that the moment of inertia tensor is symmetric. This means that the values of I_xx, I_yy, and I_zz are the same for any axis passing through the center.

I hope this helps you with your problem. Good luck!