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!

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## Comments

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TopNewestCall 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, \[\boxed{a_0=-\frac{g}{2}} \]

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Can someone please provide a link to virtual work mechanics?

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It is total work done is 0 in a fancy way

SUM Ti ai = 0

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Where can I read about principle of virtual work? Btw , very nice & smart sol.n !,+1! @Deeparaj Bhat

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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 ?

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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.

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Actually, the principle of virtual work is

farmore general. In fact, it's in the heart ofLagrangian mechanics.But, no, it's not really very related to energy conservation.

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i take it that u are invoking D'Alembert's form of the principle???

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i found this book by H.Goldstein in the basavangudi library which started with lagrangians and hamiltonian

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See this problem

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Umm. Kind of a mixture of both.

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Hope so. @Swapnil Das , @Ashish Siva please comment.!

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

I don't know any good resource.

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