Absolute motion analysis of pulley-spring system

[bremenfallturm]

Homework Statement

A mass m is suspended from the ceiling using a light spring with constant k, and two light pulleys. The mass is also attached to the floor using a light dampener, with constant C. Determine: The period time assuming undampened motion. If C>0, motivate how the period will change.

Relevant Equations

Newtons 2nd law
$$\sum \vec F = m \vec a$$

Hello!
I have this problem from an old exam I'm trying to solve. The problem is in Swedish so I've translated it:

A mass m is suspended from the ceiling using a light spring with constant k, and two light pulleys. The mass is also attached to the floor using a light dampener, with constant C. Determine: The period time assuming undampened motion. If C>0, motivate how the period will change.

NOTE that I accidentaly wrote $$C\neq 1$$ in the picture below. The correct problem statement is above.
But that part is not what I have problems with. The answer key says "if the block moves a distance x, the upper pulley moves a distance 2x, and the spring moves a distance 4x" (no further elaboration given). I can not find the last thing, aka. how the spring moves a distance 4x, even though I imagine it's very easy to figure out.
I tried solving it like this:

Forgive me in advance if I used some mechanical terms or had some grammar errors in that solution, I'm not a native English speaker.

Thank you for any help!

 

[Steve4Physics]

But that part is not what I have problems with. The answer key says "if the block moves a distance x, the upper pulley moves a distance 2x, and the spring moves a distance 4x" (no further elaboration given).

Hi. Consider the left-pulley and the rope around it.

One end of this rope is fixed to the ceiling and the other end is attached to the (centre of the) right-pulley at point P.

Q1. If the left-pulley moves down a distance x, how far does P move?

Q2. You now know how far the right-pulley moves. Applying the same reasoning as for Q1, how much is the spring stretched?

 

[bremenfallturm]

Hi. Consider the left-pulley and the rope around it.

One end of this rope is fixed to the ceiling and the other end is attached to the (centre of the) right-pulley at point P.

Q1. If the left-pulley moves down a distance x, how far does P move?

Q2. You now know how far the right-pulley moves. Applying the same reasoning as for Q1, how much is the spring stretched?

Hi, I know that the right pulley moves a distance $2x$ when the left pulley moves a distance $x$. I can come to that conclusion from the way the velocities are related, which I came up with in my solution.

I can't see how to apply the same method for the spring...

 

[haruspex]

Hi, I know that the right pulley moves a distance $2x$ when the left pulley moves a distance $x$. I can come to that conclusion from the way the velocities are related, which I came up with in my solution.

I can't see how to apply the same method for the spring...

Doesn’t pulley B have the same relationship to the spring that pulley A has to pulley B?

 

[Steve4Physics]

Hi, I know that the right pulley moves a distance $2x$ when the left pulley moves a distance $x$.

Correct. But you are not being asked how far the right pulley moves.

Say the rope is attached to the spring at point Q (end of rope). The spring is stretched by an amount equal to the distance moved by Q. You need to find how far Q moves. This is not the same as the distance moved by the right-pulley.

You didn't answer my Post #2 questions!

Q1. If the left-pulley moves down a distance x, how far does P move?

and

Q2. You now know how far the right-pulley moves. Applying the same reasoning as for Q1, how much is the spring stretched?

 

[bremenfallturm]

Sorry!

Defining point $A$ and $B$ as in my solution above ($B \iff P$ according to your definition), I get that
If the point $A$ is moved a distance $x$, point $B$ moves a distance $2x$ since $v_B=2v_A$

In my solution, I defined $s_B$ to be the distance to point $B$ from the set datum line. The length of the cord to that pulley is $l_2 = 2s_B$ and the change of $s_B$ is $v_B=2v_A$, where I'm getting confused is that doesn't the spring get stretched out with the same distance $s_B$ too?

 

[Steve4Physics]

In my solution, I defined $s_B$ to be the distance to point $B$ from the set datum line. The length of the cord to that pulley is $l_2 = 2s_B$ and the change of $s_B$ is $v_B=2v_A$, where I'm getting confused is that doesn't the spring get stretched out with the same distance $s_B$ too?

I think the problem is that you are working out $s_B$, which is the position of pulley B. But that’s not what you need to work out!

Try this...

Each pulley has it’s own rope, one end of the rope is fixed to the ceiling, the other end is 'moveable'.

We need to consider the movement of:
- each pulley;
- the moveable end of each pulley’s rope.

If a pulley moves down some distance $d$, the moveable end of its rope moves down a distance $2d$. That's the key point.

If pulley A (left) moves down a distance $d_A=x$, then the moveable end of its rope moves down a distance $2d_A = 2x$.

Pulley B (right) is attached to the moveable end of A’s rope. Therefore pulley B must move down down a distance $d_B=2x$.

***Therefore the movable end of B’s rope moves down a distance $2d_B = 2(2x) = 4x$.***

Since the moveable end of B's rope is attached to the spring, the spring is stretched by an amount $4x$.

 

[Lnewqban]

where I'm getting confused is that doesn't the spring get stretched out with the same distance $s_B$ too?

 

[Steve4Physics]

View attachment 345995

Nice idea to show what's going on! In the above attachment it might avoid possible confusion if:

- the centre of the left lever is shown moving a distance d (unmarked at present);

- the right end of the left lever is shown moving a distance 2d (marked as 'd' at present).

 

[Lnewqban]

In the above attachment it might avoid possible confusion if:

Excellent catch, Steve!
Just corrected it.
Thank you very much.

 

[bremenfallturm]

Hi! Thank you all so much for the help and intuition. What probably seems obvious and intuitive to you didn't do so to me. Additionally, my book didn't elaborate on how the rope and pulley distance change is related.
Many thanks for taking the effort to explain this using the concept of work and for making those clear pictures, they made me understand why this is the case.

Thank you a million times! I regret I found this forum now, where has it been the last 14 years or so I've spent stuyding? :P