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

  1. Feb 28, 2007 #1
    1. The problem statement, all variables and given/known data
    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).


    2. Relevant 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).

    3. 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.
     
  2. jcsd
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