Trying to find the angular speed of a rod.

[thyrgle]

Homework Statement

A uniform rod (length = 2.0m) is mounted to rotate freely about a horizontal axis that is perpendicular to the rod and that passes through the rod at a point of 0.5m from one end of the rod. If the rod is released from rest in a horizontal position, what is the angular speed of the rod as it rotates through its lowest position?

Homework Equations

I believe (although I might be wrong on some of them):
$\tau = I \alpha$
$K = \frac{1}{2} I \omega^{2}$
$L_{z} = I \omega$
For a rod the moment of inertia is:
$\frac{1}{12} M L^{2}$
And the parallel axis thereom:
$I = I_{cm} + M D^{2}$

The Attempt at a Solution

So far I have used the parallel axis theorem and the moment of inertia for a rod to get the moment of inertia to be 0.5833 for this rods rotation.
I am not sure exactly what to do after that. I can figure it out if it is the moment right after release, but not sure how to do it when the rod is at the bottom.

 

[haruspex]

Energy?

 

[thyrgle]

I first started thinking I could take the potential energy the system has before release as (mgh) and then at the bottom i could take mgh = (1/2)I*w^2 and find omega... But I couldn't figure out how to use it right...

 

[haruspex]

That's the right method. Just need to figure out h and I. Any ideas?

 

[thyrgle]

I believe I have found I for the entire rod. The issue is with the height I think. I want to say the height is 1.5 but what about the 0.5 end? I try using m = 1, g = 9.8 and h = 1.5.

1.5*9.8 = 14.7
14.7 = (1/2)(0.5833333)(w^2)
29.4 = 0.5833(w^2)
50.4 = w^2
w = 7.099

But the answer is supposedly 4.1...
I then tried subtracting the small 0.5 stub:
0.5*9.8*1 (mgh)
4.9 = (1/2)(0.5833)(w^2)
9.8 = (0.5833)w^2
16.8 = w^2
w = 4.099

Subtracting the two gives 3.00 which is not right either though.

 

[thyrgle]

$I$ was found using doing the following:

I = I_cm + m*d^2

I = (1/12)m*L^2 + m*d^2

d = 0.5 since there is a 0.5 shift from the center.

I = (1/12)*1*4 + 1*(0.5^2)

I = (0.5833333)

 

[thyrgle]

Actually never mind I should use h = 2 not 1.5 for the first part of my original attempt and then everything else should be the same.

1.5*9.8 = 14.7
14.7 = (1/2)(0.5833333)(w^2)
29.4 = 0.5833(w^2)
50.4 = w^2
w = 8.199

0.5*9.8*1 (mgh)
4.9 = (1/2)(0.5833)(w^2)
9.8 = (0.5833)w^2
16.8 = w^2
w = 4.099

8.199 - 4.099

Thanks!

Although I am still not sure why I should use 2...

 

[haruspex]

No, 0.5m is right for h (it's the distance the CoM descends).
You should have
mgh = (1/2)m(L²/12+h²)ω²
4.9*2 = (1/3+1/4)ω²
ω ≈ 4.1
Are you saying the given answer is 8.2?

 

[thyrgle]

Oh, my bad. Haha the given answer was 4.1, for some reason I didn't even realize using h = 0.5 gives the answer approx. 4.1. My bad thanks again!