This is from the solution manual of some problem of Kleppner's book. I didn't understand how the constraint equation came about to be. First of all, I don't see how that equation is equal to the length of the string. Even if it wasn't, I get why that term is constant. But so is $x_2+l_2+l'_2+x_1+l_1+l'_1$. Also, when they differentiated it, why did they neglect terms like the time derivative of $l_1$ and $l'_2$? They are obviously changing with time if $x_1$ and $x_2$ are changing with time.enter image description here

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This constraint results from having the central pulley fixed, so its height $H_o$ is fixed.

Also you can notice the height can be expressed as:

$H_o = l_1 + x_1$

regardless of how high is $M_1$.

On the other side, you can make two similar relations:

$H_o = l_1' + l_2$ following the same reasoning as before, and also one can write:

$H_o = l_1' + l_2' + x_2$ and adding the last o equations gives:

$2H_o = 2l_1' + l_2' + l_2 + x_2$ which divided by two, and added to the top equation gives:

$2H_o = l_1 + x_1 + l_1' + \frac{l_2' + l_2 + x_2}{2}$ which is a constant.

However the relation you wrote:

$l_2' + l_2 + x_2 + l_1' + l_1 + x_1$

is not a constant and this can be seen by putting the first and the thi relation, which gives:

$2H_o + l_2$

and that expression is not constant because $l_2$ can vary with time.


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