What is the Angular Speed of a Rotating Object with Varying Inertia?

AI Thread Summary
The discussion revolves around calculating the angular speed of a rotating object with varying inertia, specifically a vertical rod with an attached rectangle. The formula used for angular speed is derived from energy conservation principles, leading to the equation ω = √(2mgh/I). The moment of inertia (I) was calculated using the parallel axis theorem, with a value of 1.28. A key point raised was the correct height (h) to use in the calculation, which should be 0.625 m instead of 0.4 m. The conversation emphasizes the importance of accurately determining the center of gravity for precise calculations.
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1. A 3.0 kg vertical rod has a length of 80 cm. A 4.0 kg rectangle attached to the top of the rod has negligible dimensions. The entire object rotates counterclockwise about the bottom of the rod. Determine the angular speed of the apparatus at the instant the rod is horizontal.


2. Using energy --> mgh = .5Iωω --> ω= √(2mgh/I)3. I tried to use the parallel axis theorem to solve for I. I got 1.28. This would give me ω=√(2*7*9.8*.8/1.28) = 6.25 rad/s. I have used I= (1/12)(3)(.8*.8) + (3)(.4*.4) + (4)(.4*.4) plus several different variations.
 

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Where is the effective mass of the rod? How far does it fall?
 
I think I have I have it figured out - I needed to use the center of gravity for my height in the mgh part of the problem --> so h = .625m not .4m, which is what I was doing before. Thanks!
 
Are you sure it's 0.625 m?
 

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