Using inertia tensor in conservation of angular momentum

[Mark24]

Hi all, this is my first post. I'm under the gun at the moment because we were just told yesterday to expect a similar problem to the one below on our final, yet we were not taught how to work such a problem in class. (Our teacher does not write the final hence the discrepancy.)

I have a solution manual I am trying to learn how to do this problem with the help of, but it is not going very well. My book only vaguely touches on the inertia tensor so I'm very confused about what's going on in this problem. I've posted the two pages from the solution manual. I keep trying but I can't seem to duplicate the numbers the solution manual gets.

1)The problem statement, all variables and equations:
Page 1: http://i43.tinypic.com/2j4d761.jpg
Page 2: http://i41.tinypic.com/w1mhg.jpg

2) Attempt at solution
The 'I1' calculation seems to be I_zz for a thin plate from the back of my book, where I_zz = 1/12*m(a² + b²). Yet the manual labels that value I1 and seems to use it in place of where I_xx should be in the inertia tensor (I_mat.) I don't know if this is an error or not.

And also for I_oa, I'm using the formula I_xx*u_x² + I_yy*u_y² + I_zz*u_z² where the u's are the unit vectors for r_o/a. Since u_x is zero, I should get: I_oa = I_yy*u_y² + I_zz*u_z². Again, I'm unsure of which is I_yy and I_zz from the solution manual due to the obscure labeling of I1/I2/I3, and I'm unsure of how to identify which is which from the picture. (that's something I never figured out since we are usually given the values)

If anyone can offer some tips or suggestions on this problem, I would be very grateful. After spending about 4 hours staring at it, googling, etc. I'm really at my wits end.

 

[tiny-tim]

Welcome to PF!

The 'I1' calculation seems to be I_zz for a thin plate from the back of my book, where I_zz = 1/12*m(a² + b²). Yet the manual labels that value I1 and seems to use it in place of where I_xx should be in the inertia tensor (I_mat.) I don't know if this is an error or not.

Hi Mark24! Welcome to PF!

The 1 2 and 3 refer to x y and z, so the I₁ is I_xx, which they get by adding I_yy and I_zz (the parallel axis theorem).

I don't understand why you're confused … 1,2,3 ~ x,y,z is fairly simple notation

Anyway, look at the diagram …​

I_zz clearly depends on a, and it's through the c.o.m., so it's small, so it's 1/12

while I_yy clearly depends on b, and it's not through the c.o.m., so it's larger, so it's 1/3

 

[Mark24]

Thanks tiny-tim, how they derived the moments of inertia makes sense now after you explained it. Since the inertias seem to be correct but I'm still getting the wrong answer, I must be messing up the final equation - the one listed in the second picture as:
I_mat(0,0,w₁)oa = I_oa*w₂

For I_oa, I get I_oa = I_yy*u_y² + I_zz*u_z² = .192

For I_mat, I get a diagonal matrix with values: .4167, .2667, .15. w₁ is 2 rad/s.

If I do I_mat*(0,0,w₁) = I_oa*w₂, I get w₂ = 1.56 in the z direction and 0 in the y, which is wrong. Any idea what I am missing? The solution has an 'oa' at the end on the left hand side of the equation. oa is the unit vector, so I don't understand why it would be needed in the cons. of angular momentum equation.

 

[tiny-tim]

Hi Mark24!

(have an omega: ω )

If I do I_mat*(0,0,w₁) = I_oa*w₂, I get w₂ = 1.56 in the z direction and 0 in the y, which is wrong. Any idea what I am missing? The solution has an 'oa' at the end on the left hand side of the equation. oa is the unit vector, so I don't understand why it would be needed in the cons. of angular momentum equation.

I_matrix converts the angular velocity vector (0,0,ω₁) to the angular momentum vector, I_mat*(0,0,ω₁).

The external forces are at O and A, so angular momentum is only conserved about the OA axis, so we need the OA-component of the angular momentum vector:

I_mat*(0,0,ω₁) "dot" oa