my attempt
$$4(x_3-c_1)+x_p=L$$
which give $4*a_3=-a_p$
such that $a_p$ acceleration of pulley
how to get relation between $a_1$ and $a_2$ and $a_3$ using conversion of string and how to solve the same problem by work done by tension
Why is the net work done in a pulley-string system zero?
Best Answer
Instead of thinking about accelerations $a_1$, $a_2$, and $a_3$, it's easier to think about displacements $x_1$, $x_2$, and $x_3$. Furthermore, imagine replacing the string with a rubber band, which can freely stretch or shrink, so you can imagine one mass moving at a time.
Given a displacement $x_1$, the rubber band shrinks by $2x_1$. Similarly, given a displacement $x_2$, the rubber band shrinks by $2 x_2$, and given displacement $x_3$ it shrinks by $x_3$. But since we actually have a string, the total length of the string must stay constant, so $$2x_1 + 2x_2 + x_3 = 0.$$ By differentiating both sides with respect to time twice, we have $$2a_1 + 2a_2 + a_3 = 0.$$ This is the desired relation between the accelerations, using conservation of string.