String conservation with springs

Question

I was trying to solve this Problem that asked to find the period of small oscillations for this system.

To do so I used the fact that for a massless pulley with strings around it, the sum of the displacements of the string around the pulley is twice the displacement of the pulley.

This gives the correct answer but it bothered me because the above fact is dependent on the length of the string being constant but springs can change their length.

Now:

1. Can I justify this by saying that for small oscillations the change in string length is approximately zero?
2. If not, then how can I find the period otherwise?

Answer 1

For problems like this one, it is generally assumed that the string is inextensible, meaning that it's length is constant(usually given in problem statement). So, the elongation comes only from the extension of the springs and not the string.

Hope this helps!