Some comments on modeling:
- Since we are looking at rotation about the trunk of the body, the best views are the TOP views shown above.
- As written in the problem statement, the torso is to modeled as a circular cylinder with a weight of 135 pounds and radius of 6.5 inches. Recall that the centroidal mass moment of inertia (MMI) of a cylinder is (1/2)*m*r^2. This MMI should be used for the torso only in the initial state.
- Also as written, the arms are to be THIN rods. In the initial state, the arms appear as thin bars that are pinned at a distance of 6.5 inches from the rotation axis of the trunk (see TOP view above). Do not forget to use the parallel axis theorem to find the MMI of each arm about the rotation axis of the trunk.
- In the final state, the problem statement says to lump together the torso and arms in a single homogeneous cylinder with a weight of 165 pounds and radius of 6.5 inches.
7 comments:
Can you please clarify a little more on how to find the MMI for the final and initial states with the arms extended and down to the side?
I got
MMI(1)=(2/3)*Marm*l^2+2*Marm*d1^2+(1/2)*Mcyl*r^2
MMI(2)=.5*Mcyl*r^2+2*Marm*d2^2
but my answer ended up being 4.505. Does anyone see what is wrong with my MMI's.
I used H01=H02e
“In the final state, the arms appear as "point masses" at a distance of 6.5 inches from the rotation axis of the trunk…”
I believe the book tells us to treat the man as a solid 165 lb cylinder for the final state, correct?
I stand corrected on this (thanks to tjmackey for pointing out this). I have modified my original post to reflect this correction.
Sorry for the confusion on this. I need to read the questions more carefully ...
"Treat the man with arms retracted as a solid 165-lb cylinder of 13-in. diameter."
So, I used
(1/2)*(165/32.2)*(6.5/12)^2
as the MMI for the final state.
I'm having trouble with my MMI for the initial state. I used
(1/3)*m _torso*(r)^2 +
2[(1/3)*m_arm*(l)^2 + m_arm*(r)^2]
where r = 6.5/12ft and l = 27/12ft
I can't get the right answer.
Any suggestions?
-John
John,
I see a couple things in your MMI for the initial state:
* Your first term (1/3)*m _torso*(r)^2 should be (1/2)*m _torso*(r)^2. [I suspect that was a typo on your post.]
* The other is more subtle. In the second term, you used the PAT to move the MMI from the END of the arm to the center of the torso. The PAT applies only for going directly from the CM of the arm to the center of the torso. The second term should read: 2[(1/12)*m_arm*(L)^2 + m_arm*(r+L/2)^2] where L is the length of the arm. [You might think that your and my expressions are the same, but they aren't. Let me know if you do not see how they differ, or if you are not sure of the application of the PAT.]
CMK
Thank you!
* Yes, that was a typo.
* That fixed it. I see that I used the PAT twice, as it was already in the formula I picked to use the PAT on.
-John
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