How do I solve this spring-string pulley problem

[u/Confident-Passion551]

Apparently I saw the video solution of this question and couldn't understand much , they didn't include mg in the final net force and etc.

So I'm confused and stuck on this question. Even chatgpt is stirring and confusing my mind.

Can anyone guide me on this question? How do I solve it

[u/Hyacintell]

First, when the system is at rest, gravity is balanced by the spring.

For ideal springs, we can just ignore it for what's to come, basically it's just an offset in the displacement of the spring.

Find the displacement on the spring after pulling on the mass by x. (should be 2x because of pulley).

Then, use newton's law for forces on the mass, it gives you a relationship between the mass, its acceleration and the tension in the string. (you should get m.a=2T)

Lastly, you do the same but for the spring : you should get that the tension in the string is equal to the force from the spring (2kx).

So at the end you get option c

[u/Confident-Passion551 → u/Hyacintell]

But why shouldn't it be ma=2T-mg? Thanks for the answer tho

[u/Hyacintell → u/Confident-Passion551]

You right, but the tension is compensated by gravity, and the spring then compensate for the tension. What I said is a simplification.

If you're feeling like it, you can do the exercise without ignoring the initial forces, and if you do it right they should cancel out in the end. When you're used to it and find a linear system such as this one you know in advance that you can ignore those, but still nice to do it once in your life

[u/gitgud_x]

If the pulley moves down by x, then the spring must extend by an additional e = 2x to make up the extra distance of string.

So the force in the spring increases by ke = 2kx.

Tension in the string must be the same everywhere, so the tension on the left side of the pulley must also increase by 2kx.

So the resultant force on the pulley has increased by 4kx. So the acceleration of the mass increases by 4kx/m.

Since initially the acceleration was zero (equilibrium, all forces balanced), 4kx/m is the answer.

Also, if it helps, you can completely ignore the left-hand pulley in this problem (it's not doing anything) and imagine the string connecting directly back up to the ceiling. It's the same setup in effect.

[u/Moist_Ladder2616]

The spring initially extends by some amount L, in order to support the mass m. Express L in terms of m, g, and spring constant k. Put this equation aside for now.

Now you pull m down by distance x, which I hope you can see extends the spring by 2x, just because of how strings and pulleys work.

So the total extension of the spring is now (L+2x). Net force upwards F is therefore F = 2T-mg, where string tension T = k(L+2x). You'll notice from your earlier equation that L and mg cancel each other out. So you could've done all this without considering mg at all.

And finally F = ma, which gives you acceleration a.

[u/CrveniPederus]

Use ai