Inertia matrix and centre of gravity of a set of rods

[Bucky]

Homework Statement

Three uniform rods OA, OB, and OC, each of mass 3m and length a, lie along the x, y and z axes, respectivley. Show that the inertia matrix relative to Oxyz is

$Io = 2ma^2 \left(\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right)$
Determine the inertia matrix with respect to a parallel frame through the centre of mass G.

Homework Equations

The Attempt at a Solution

The proof was straightforward enough, but finding Ig has proven difficult. The book lists it as
$Ig = 1/4ma^2 \left(\begin{array}{ccc}6&-1&-1\\-1&6&-1\\-1&-1&6\end{array}\right)$

Our current problem is finding the centre of mass of the system. I thought it would be G(a/2, a/2, a/2), as each rod is of length a, but the answer lists this as G(a/6, a/6, a/6). Can anyone point out where I've gone wrong? Thanks.

 

[mighty2000]

Hello,

It seems like you are on the right track with your calculations. However, there is a small mistake in your calculation of the centre of mass. The correct coordinates for the centre of mass G are (a/6, a/6, a/6), as listed in the solution.

To find the centre of mass of a system, we use the formula:

x_cm = (m1*x1 + m2*x2 + m3*x3)/M_total

where m1, m2, m3 are the masses of the three rods and x1, x2, x3 are the coordinates of their respective centres of mass. M_total is the total mass of the system, which in this case is 9m.

Using this formula, we get:

x_cm = (3m*0 + 3m*a + 3m*0)/(9m) = a/3

Similarly, for the y and z coordinates, we get y_cm = a/3 and z_cm = a/3.

Therefore, the centre of mass of the system is (a/3, a/3, a/3). To find the coordinates with respect to the origin O, we need to subtract a/3 from each coordinate, which gives us (a/6, a/6, a/6).

Substituting these values into the formula for the inertia matrix with respect to the centre of mass, we get:

Ig = 1/4ma^2 \left(\begin{array}{ccc}6&-1&-1\\-1&6&-1\\-1&-1&6\end{array}\right)

I hope this helps clarify your confusion. Let me know if you have any further questions. Keep up the good work!