Undergrad Block sliding down an incline plane

[VVS2000]

TL;DR

In the given figure, if the block slides down by some x metres, does the spring also get strecthed by x metres? Or will it get strecthed due to the horizontal component of the force acting on the sliding block? Hence the expansion in the spring is caused due to the horizontal component of the force

 

[anuttarasammyak]

Summary:: In the given figure, if the block slides down by some x metres, does the spring also get strecthed by x metres?

Yes, it does. And the direction of the force pulling the wire is along the slope.

 

[VVS2000]

Yes, it does.

Any hint on how to arrive at that equality? Like a mathematical proof?

 

[anuttarasammyak]

You see in the figure, say x stretched the spring, same x the bock goes along the slope.

 

[Lnewqban]

That pulley only changes the direction of the tension force in the string or rope that connects the spring and the sliding block.

Because of that, you can create two different coordinate x-y systems: one for the FBD of the horizontal spring, in which the x-axis is horizontal and aligned with it, and another for the sliding block, in which the x-y system is inclined, being the x-axis aligned with the slope.

For each FBD, you will have a force T of the same magnitude (mgcos<).
If the angle of the slope increases, the value of force T acting on the spring also increases, until reaching the maximum value of mg at angle 90 degrees.

Please, see examples that are shown in this link:
https://courses.lumenlearning.com/suny-osuniversityphysics/chapter/5-7-drawing-free-body-diagrams/

:)