# 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 years, 7 months 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, $\boxed{a_0=-\frac{g}{2}}$

- 3 years, 7 months ago

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

- 3 years, 7 months 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 right @Deeparaj Bhat ?

- 3 years, 7 months 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.

- 3 years, 7 months ago

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.

- 3 years, 7 months ago

Yeah, but I have used the latter term often.

- 3 years, 7 months ago

yeah!! its more related to least action

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

- 3 years, 7 months ago

Yeah.

- 3 years, 7 months ago

Why do you know lagrangian mechanics?:p

- 3 years, 7 months ago

lol

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

- 3 years, 7 months 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?

- 3 years, 7 months ago

That book was put in general by Julien in slack.

- 3 years, 7 months ago

yes i know. that's how i got it...

- 3 years, 7 months ago

Same here.

- 3 years, 7 months ago

yes :)

- 3 years, 7 months 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 energy

- 3 years, 7 months 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.

- 3 years, 7 months ago

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

- 3 years, 7 months ago

Are you all college guys!!

- 3 years, 7 months ago

Nah I am in 11th :P

- 3 years, 7 months ago

Just finished 12th :P

- 3 years, 7 months ago

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

- 3 years, 7 months ago

Yes both are there.

- 3 years, 7 months ago

Umm. Kind of a mixture of both.

- 3 years, 7 months ago

See this problem

- 3 years, 7 months ago

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

I don't know any good resource.

- 3 years, 7 months ago

- 3 years, 7 months ago

It is total work done is 0 in a fancy way

SUM Ti ai = 0

- 2 years, 1 month ago