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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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, \[\boxed{a_0=-\frac{g}{2}} \] – Deeparaj Bhat · 10 months ago

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Can someone please provide a link to virtual work mechanics? – Rishabh Tiwari · 9 months, 4 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 · 10 months ago

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@Deeparaj Bhat ? – Anirudh Chandramouli · 9 months, 4 weeks ago

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 rightLog in to reply

– Ashish Siva · 9 months, 4 weeks ago

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.Log in to reply

farmore general. In fact, it's in the heart ofLagrangian mechanics.But, no, it's not really very related to energy conservation. – Deeparaj Bhat · 9 months, 4 weeks ago

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– Faraz Khan · 9 months, 3 weeks ago

Are you all college guys!!Log in to reply

– Deeparaj Bhat · 9 months, 3 weeks ago

Just finished 12th :PLog in to reply

– Ashish Siva · 9 months, 3 weeks ago

Nah I am in 11th :PLog in to reply

– Anirudh Chandramouli · 9 months, 3 weeks ago

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 energyLog in to reply

– Deeparaj Bhat · 9 months, 3 weeks ago

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.Log in to reply

– Anirudh Chandramouli · 9 months, 3 weeks ago

ok thanks anyways. I am new to Lagrangian mechanics and hope I will master it soon. :)Log in to reply

i take it that u are invoking D'Alembert's form of the principle??? – Rohith M.Athreya · 9 months, 4 weeks ago

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– Anirudh Chandramouli · 9 months, 3 weeks ago

Why do you know lagrangian mechanics?:pLog in to reply

i found this book by H.Goldstein in the basavangudi library which started with lagrangians and hamiltonian – Rohith M.Athreya · 9 months, 3 weeks ago

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– Anirudh Chandramouli · 9 months, 3 weeks ago

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?Log in to reply

– Rohith M.Athreya · 9 months, 3 weeks ago

yes :)Log in to reply

– Deeparaj Bhat · 9 months, 3 weeks ago

That book was put in general by Julien in slack.Log in to reply

– Anirudh Chandramouli · 9 months, 3 weeks ago

yes i know. that's how i got it...Log in to reply

– Deeparaj Bhat · 9 months, 3 weeks ago

Same here.Log in to reply

– Deeparaj Bhat · 9 months, 4 weeks ago

Yeah.Log in to reply

– Ashish Siva · 9 months, 4 weeks ago

Yeah, but I have used the latter term often.Log in to reply

this problem – Rishabh Tiwari · 9 months, 4 weeks ago

SeeLog in to reply

– Swapnil Das · 9 months, 4 weeks ago

Umm. Kind of a mixture of both.Log in to reply

@Swapnil Das , @Ashish Siva please comment.! – Rishabh Tiwari · 9 months, 4 weeks ago

Hope so.Log in to reply

– Ashish Siva · 9 months, 4 weeks ago

Yes both are there.Log in to reply

I don't know any good resource. – Deeparaj Bhat · 9 months, 4 weeks ago

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