# Ambiguous Constraint equation

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.

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.