Acceleration of the midpoint of a light rod

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The discussion revolves around the calculation of the acceleration of the midpoint of a light rod under the influence of a force. Initially, the calculations assumed a rod length of 6l and a center of mass located at 2l from a 2m mass, leading to an incorrect acceleration result. The correct approach involves recognizing that the mass in the diagram is actually 3m, not 2m, which aligns the computed acceleration with the expected answer of F/3m. There is also a debate about whether the cords are inextensible, affecting the interpretation of forces in the system. Ultimately, the confusion stemmed from misinterpreting the mass values in the problem setup.
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I assumed the length of the rod to be ##6l## for simpler calculations . Here , $$a_{cm}=\frac{F}{m+2m}=\frac{F}{3m}$$
Here , CM is located at a distance of ##2l## from the ##2m## mass . Writing the moment equation : $$\alpha \times \left[ m(4l)^2+2m(2l)^2\right]=Fl\implies \alpha=\frac{F}{24ml}\implies \alpha l=\frac{F}{24m}$$ $$a_{\text{mid}} = a_{CM}+\alpha l=\frac{F}{3m}+\frac{F}{24m}=\frac{3F}{8m}$$ which doesn't match the answer key .
 
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If the rod is spinning about its center of mass, replace the masses with the tension developed in each bar (they say inextensible cords - but I don't think that is consistent ) in the FBD of the rod, and you should be using the thin rods moment of inertial about the center of mass then apply ## \sum \tau = I_{cm} \alpha ##

Scratch that, massless rod. Looks fine to me. Did you notice they put a mass of ##3m## in the diagram?

Another thing, maybe they do actually mean cords, i.e. they don't accept compression? It wouldn't hurt if you gave the answer they expect.
 
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My bad , the answer matches if I took the mass as given in the diagram , i.e, ##3m## instead of ##2m## . The answer turns out to be : ##\frac{F}{3m}## .
 
Bling Fizikst said:
My bad , the answer matches if I took the mass as given in the diagram , i.e, ##3m## instead of #2m## . The answer turns out to be : ##\frac{F}{3m}## .
That's not your bad, it's their bad!
 
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