Spring attached to two hanging objects

[GoGoGadget]

Homework Statement

Draw a picture of a spring connected over pulleys to two hanging objects of equal
weight at equilibrium. Determine the relationship of forces among each individual object, then find a final equation that describes the overall relationship of the objects to one another.

Homework Equations

F = ma

Hooke's law for springs:
F_s = -kx

The Attempt at a Solution

Body-diagram of the model described above is attached to this post. I know the equation for describing the force of a spring is by Hooke's Law:

F_s = -kx

Forces acting on just the spring (for both directions):

F_y = ma_y = 0

F_x = T-kx = ma_x

Forces acting on M1 & M2:

F_x = ma_x = 0

F_y = T-mg = ma_y = 0

T-mg = 0

T = mg

Relationship of spring with M1 & M2:

So if T = mg, we can substitute T for mg in the equation. I know we need to subtitute the equation so the relationship can be determined on one component and for this, I did it for the x axis. Here's what I have so far:

F_x = ma_x

-kx-M1-M2=ma_x

From here, I'm not sure what to do. On the one hand, I know that the masses on both ends of the spring on going to keep extending the spring out. And I know that the masses are going to be proportional to one another as the spring continues to extend. However, I'm not sure what to make of the ma_x portion of the equation. I know that M1 and M2 are equivalent to the force of mg but when the m in the mz_x is divided, that would cancel out the "m" variables of M1 and M2. Any help from here is appreciated.

 

[szimmy]

F_s = -kx

Forces acting on just the spring (for both directions):

F_y = ma_y = 0

F_x = T-kx = ma_x

Forces acting on M1 & M2:

F_x = ma_x = 0

F_y = T-mg = ma_y = 0

T-mg = 0

T = mg

Relationship of spring with M1 & M2:

So if T = mg, we can substitute T for mg in the equation. I know we need to subtitute the equation so the relationship can be determined on one component and for this, I did it for the x axis. Here's what I have so far:

F_x = ma_x

-kx-M1-M2=ma_x

From here, I'm not sure what to do. On the one hand, I know that the masses on both ends of the spring on going to keep extending the spring out. And I know that the masses are going to be proportional to one another as the spring continues to extend. However, I'm not sure what to make of the ma_x portion of the equation. I know that M1 and M2 are equivalent to the force of mg but when the m in the mz_x is divided, that would cancel out the "m" variables of M1 and M2. Any help from here is appreciated.

The force you're calling tension, you should be calling something else. What other force is acting on the masses aside from gravity?

You saying Fy = ma_y = 0 is correct, but what are the forces that cancel out so that happens? Expand your Fy equation.

"However, I'm not sure what to make of the ma_x portion of the equation."
You stated before that the objects were in equilibrium, so isn't ma_x = 0?

 

[GoGoGadget]

The force you're calling tension, you should be calling something else. What other force is acting on the masses aside from gravity?

You saying Fy = ma_y = 0 is correct, but what are the forces that cancel out so that happens? Expand your Fy equation.

"However, I'm not sure what to make of the ma_x portion of the equation."
You stated before that the objects were in equilibrium, so isn't ma_x = 0?

Sorry, I probably should have been more clear in my confusion. I'm trying to figure an equation for the relationship between the two hanging mass objects from the spring and showing how they impact the extension of the spring. I attempted before to show it on one axix (the x-axis) but wasn't sure if it I had been on the right track or not with it.

 

[haruspex]

-kx-M1-M2=ma_x

I don't know what you mean by that equation. From the rest of the post, M1 and M2 are labels, not masses or forces. Each has mass m. What is a_x the acceleration of?

 

[GoGoGadget]

I was trying to figure out or understand if there can be a possible "acceleration" for when the hanging weights from either end of the spring in causing it to extend out? This was a general problem given that we had to solve from my lab manual as pre-lab question and I guess I didn't understand as to whether or not a possible acceleration existed. From the diagram, M1 and M2 are the hanging weights on either end of the spring and I know from this, each mass would have a force of weight (w) which is equivalent to their mass sizes and gravity (m*g). Is there an acceleration for the hanging weights on both ends of the spring in terms of the spring's extension?

 

[haruspex]

To be honest, I'm not sure what the question is asking for. It says equilibrium, so there's no acceleration. You can write one equation for each mass and one for the spring, but I don't know what they want as one equation to encapsulate everything. I can think of a couple of possibilities, but they're rather artificial. Maybe they just mean an equation relating spring extension to the masses and gravity.
Anyway, you didn't answer my questions: what do M1 and M2 stand for in that equation, and how are you defining a_x?

 

[GoGoGadget]

To be honest, I'm not sure what the question is asking for. It says equilibrium, so there's no acceleration. You can write one equation for each mass and one for the spring, but I don't know what they want as one equation to encapsulate everything. I can think of a couple of possibilities, but they're rather artificial. Maybe they just mean an equation relating spring extension to the masses and gravity.
Anyway, you didn't answer my questions: what do M1 and M2 stand for in that equation, and how are you defining a_x?

M1 and M2 stand for m1g and m2g in my equation. For the x axis, I am defining it along the horizontal axis to incorporate the force of the spring in the equation. I restated what I originally defined as tension from the pull of the string to just the force of the pull of the string (F_string). In going back to look over the problem again, to show F_y = ma_y = 0 along the y axis, I made m1g and m2g equivalent to F_string but I don't know if this is correct or not. From there, it would make sense to show an equation that relates spring extension to the masses and gravity but I'm not sure if I'm on the right track with that or not. Thanks so much for your help with this problem!

 

[haruspex]

M1 and M2 stand for m1g and m2g in my equation. For the x axis, I am defining it along the horizontal axis to incorporate the force of the spring in the equation. I restated what I originally defined as tension from the pull of the string to just the force of the pull of the string (F_string). In going back to look over the problem again, to show F_y = ma_y = 0 along the y axis, I made m1g and m2g equivalent to F_string but I don't know if this is correct or not. From there, it would make sense to show an equation that relates spring extension to the masses and gravity but I'm not sure if I'm on the right track with that or not. Thanks so much for your help with this problem!

Yes, that's the right track, but given your explanations the equation you had, -kx-M1-M2=ma_x, is definitely wrong.
In the OP, you correctly wrote, for the masses: F_y = T-mg = ma_y = 0
For the spring, you wrote
F_y = ma_y = 0
F_x = T-kx = ma_x
What's m there - the mass of the spring? It's very confusing when you use the same symbol for different things. OK, you've made the context clear in each case, so no confusion so far, but at some point you will combine equations from different contexts, and then it becomes impossible to follow.
Does the spring have mass of any significance? Anyway, a_x is again 0, right? So T = kx.