GR0177 #91



Alternate Solutions 
Ning Bao 20080201 08:11:02  Let R, m,h,g=1. Then you just have
1=1/2*8/7+1/2*I*8/7
1=4/7+4/7*I
I=3/4.  

Comments 
redmomatt 20111008 15:23:01  Did this by process of elimination (not even looking at the PE versus KE).
(A) can't be true, since the cylinder is not of uniform density.
(D) can't be true, since this would only occur if all of the mass were concentrated in a thin ring at a distance .
(E) can't be true, since the largest moment would be (D).
We are left with (B) and (C). Because a radial mass distribution that varies as should be less than something very close to (D), we are left with choice (B).
Nezumi 20120209 18:27:24 
The radial dependence is not given to be linear. I Think (B) or (C) remain until you look at energy.

Nezumi 20120306 12:06:24 
I was looking at the problem again. The elimination you did is the right place to start. Once you have it down to (B) and (C), look at the translational energy.
since the translation takes up a little more than half of the original energy of the object, less than half is available for the rotational energy. This implies a smaller moment of inertia, hence choice (B)

  Ning Bao 20080201 08:11:02  Let R, m,h,g=1. Then you just have
1=1/2*8/7+1/2*I*8/7
1=4/7+4/7*I
I=3/4.
smokwzbroiplytowej 20081021 11:27:20 
Nice  sure helps save time! Caveat: you really can't set and simultaneously, since they are both measures of length.
Here it works rather nicely, because H cancels out as you could tell from the answers.

  neutrino 20071031 03:28:11  You can make the calculation more easy by noting that the rotational inertia will look like
,
with some constant. Now, you just have to solve:
This might save you some time. Because, 1.7 min..., man, that is not much time.
tommy22222 20191022 14:57:59 
BRILLIANT!!! I think this is the best and the most make sense alternative solution under this pose.

  Richard 20071030 18:38:31  A potentially useless note:
The solution given in (E) is found by NOT accounting for the translational kinetic energy.
  Andresito 20060305 00:00:42  Yosun, there is a typo on your solution. When you express I, the term MV^2 has a factor of 2 implicitly.
Recall the conservation of energy equation, multiply both terms by 2 and the "only" term that has now a factor of 2 is MgH.
The correct expression should have I=(2 MgH  MV^2) (r/v)^2
Yosun, thank you very much for working out the problems. If you are a girl I send you a big kiss :). I obtained the pdf from the yahoogroup. It also has the typo.   keflavich 20051110 22:48:10  You don't need to consider as part of the gravitational potential energy. When the cylinder reaches the bottom of the plane, its center of mass will remain at a distance R above the ground.
yosun 20051110 22:54:19 
keflavich: you're right that the center of mass stays at a height of R above the ground at both points. thanks for the correction.

 

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