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The Infinite Atwood Machine

Consider the infinite Atwood’s machine shown in the figure.

A string passes over each pulley, with one end attached to a mass and the other end attached to another pulley. All the masses are equal to m, and all the pulleys and strings are massless. The masses are held fixed and then simultaneously released. What is the acceleration of the top mass?

I have given the problem statement as it is. My humble request is for you all to help me obtain a mathematical solution to the above question rather than a trivial physical interpretation. Cheers!

Note by Anirudh Chandramouli
3 months, 4 weeks ago

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Call the topmost block as \(Block_0\), the next block as \(Block_1\) and so on.

Let \(a_i\) denote the acceleration (in the downward direction) of \(Block_i\) for \(i=0,1,2 \ldots\).

Then, by the principle of virtual work, we have: \[\sum_{i=0}^{\infty} \frac{a_i}{2^i} = 0 \]

By Newton's second law, \[mg - \frac{T}{2^i} = ma_i \quad i=0,1,2 \ldots \] Where \(T\) is the tension in the string connecting \(Block_0\).

Now, plugging in the value of \(a_i\) (from the second equation), into the first one, and some simplification, we get \[T = \frac{3}{2} mg \]

Hence, \[\boxed{a_0=-\frac{g}{2}} \] Deeparaj Bhat · 3 months, 4 weeks ago

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Can someone please provide a link to virtual work mechanics? Rishabh Tiwari · 3 months, 3 weeks ago

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Where can I read about principle of virtual work? Btw , very nice & smart sol.n !,+1! @Deeparaj Bhat Rishabh Tiwari · 3 months, 4 weeks ago

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@Rishabh Tiwari I think the general notion of conservation of string is termed as principle of virtual work by some other concept of energy conservation, maybe? Am I right @Deeparaj Bhat ? Anirudh Chandramouli · 3 months, 3 weeks ago

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@Anirudh Chandramouli I believe in the latter. It is more commonly used than conplex words like "virtual work". Virtual work is nice but in real life it does not really apply because no machine is 100% efficient. Conservation of energy is better because it even covers dissipated energy. Ashish Siva · 3 months, 3 weeks ago

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@Anirudh Chandramouli Actually, the principle of virtual work is far more general. In fact, it's in the heart of Lagrangian mechanics.

But, no, it's not really very related to energy conservation. Deeparaj Bhat · 3 months, 3 weeks ago

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@Deeparaj Bhat Are you all college guys!! Faraz Khan · 3 months, 3 weeks ago

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@Faraz Khan Just finished 12th :P Deeparaj Bhat · 3 months, 3 weeks ago

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@Faraz Khan Nah I am in 11th :P Ashish Siva · 3 months, 3 weeks ago

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@Deeparaj Bhat But isn't the basis for Lagrangian mechanics that Euler Lagrangian formula for energy or whatever. That is why I assumed that it had to do with energy Anirudh Chandramouli · 3 months, 3 weeks ago

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@Anirudh Chandramouli It more or less comes to that. But, it kicks off with virtual work. After a lot of simplification (and assumptions), you get the form that you're talking about. Deeparaj Bhat · 3 months, 3 weeks ago

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@Deeparaj Bhat ok thanks anyways. I am new to Lagrangian mechanics and hope I will master it soon. :) Anirudh Chandramouli · 3 months, 3 weeks ago

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@Deeparaj Bhat yeah!! its more related to least action

i take it that u are invoking D'Alembert's form of the principle??? Rohith M.Athreya · 3 months, 3 weeks ago

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@Rohith M.Athreya Why do you know lagrangian mechanics?:p Anirudh Chandramouli · 3 months, 3 weeks ago

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@Anirudh Chandramouli lol

i found this book by H.Goldstein in the basavangudi library which started with lagrangians and hamiltonian Rohith M.Athreya · 3 months, 3 weeks ago

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@Rohith M.Athreya Oh I'm reading that book as well. What a pleasant coincidence. Is it the one that starts with the mechanics of a single particle and then goes on to systems of particles and stuff? Anirudh Chandramouli · 3 months, 3 weeks ago

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@Anirudh Chandramouli yes :) Rohith M.Athreya · 3 months, 3 weeks ago

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@Anirudh Chandramouli That book was put in general by Julien in slack. Deeparaj Bhat · 3 months, 3 weeks ago

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@Deeparaj Bhat yes i know. that's how i got it... Anirudh Chandramouli · 3 months, 3 weeks ago

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@Anirudh Chandramouli Same here. Deeparaj Bhat · 3 months, 3 weeks ago

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@Rohith M.Athreya Yeah. Deeparaj Bhat · 3 months, 3 weeks ago

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@Deeparaj Bhat Yeah, but I have used the latter term often. Ashish Siva · 3 months, 3 weeks ago

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@Anirudh Chandramouli See this problem Rishabh Tiwari · 3 months, 3 weeks ago

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@Anirudh Chandramouli Umm. Kind of a mixture of both. Swapnil Das · 3 months, 3 weeks ago

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@Anirudh Chandramouli Hope so. @Swapnil Das , @Ashish Siva please comment.! Rishabh Tiwari · 3 months, 3 weeks ago

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@Rishabh Tiwari Yes both are there. Ashish Siva · 3 months, 3 weeks ago

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@Rishabh Tiwari I learnt from an old book of my dad's.

I don't know any good resource. Deeparaj Bhat · 3 months, 3 weeks ago

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