Absolute motion analysis of pulley-spring system

In summary, the analysis of a pulley-spring system in absolute motion examines the dynamics of the system by considering the forces acting on the pulley and the spring. This involves calculating the tension in the spring and the acceleration of the masses involved, while taking into account the constraints imposed by the pulley. The study aims to derive equations of motion that describe the system's behavior under various conditions, allowing for predictions of its response to external forces and changes in configuration. This analysis is crucial for applications in mechanical engineering and physics, where understanding the motion of interconnected components is essential for design and functionality.
  • #1
bremenfallturm
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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
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 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:
1716628415039.png

1716628429003.png


1716628441913.png

1716628459857.png

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!
 
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  • #2
bremenfallturm said:
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?
 
  • #3
Steve4Physics said:
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 when the left pulley moves a distance . 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...
 
  • #4
bremenfallturm said:
Hi, I know that the right pulley moves a distance when the left pulley moves a distance . 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?
 
  • #5
bremenfallturm said:
Hi, I know that the right pulley moves a distance when the left pulley moves a distance .
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!

Steve4Physics said:
Q1. If the left-pulley moves down a distance x, how far does P move?
and
Steve4Physics said:
Q2. You now know how far the right-pulley moves. Applying the same reasoning as for Q1, how much is the spring stretched?
 
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  • #6
Sorry!

Defining point and as in my solution above ( according to your definition), I get that
If the point is moved a distance , point moves a distance since

In my solution, I defined to be the distance to point from the set datum line. The length of the cord to that pulley is and the change of is , where I'm getting confused is that doesn't the spring get stretched out with the same distance too?
 
  • #7
bremenfallturm said:
In my solution, I defined to be the distance to point from the set datum line. The length of the cord to that pulley is and the change of is , where I'm getting confused is that doesn't the spring get stretched out with the same distance too?
I think the problem is that you are working out , 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 , the moveable end of its rope moves down a distance . That's the key point.

If pulley A (left) moves down a distance , then the moveable end of its rope moves down a distance .

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

***Therefore the movable end of B’s rope moves down a distance .***

Since the moveable end of B's rope is attached to the spring, the spring is stretched by an amount .
 
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  • #8
bremenfallturm said:
where I'm getting confused is that doesn't the spring get stretched out with the same distance too?
Pulleys in series 1.jpg


Pulleys in series 2.jpg


Pulleys in series 3.jpg
 

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Last edited:
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  • #9
Lnewqban said:
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).
 
  • #10
Steve4Physics said:
In the above attachment it might avoid possible confusion if:
Excellent catch, Steve!
Just corrected it.
Thank you very much.
 
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  • #11
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
 
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FAQ: Absolute motion analysis of pulley-spring system

What is absolute motion analysis in a pulley-spring system?

Absolute motion analysis refers to the study of the motion of objects in a pulley-spring system without considering any relative motion. It focuses on the actual positions, velocities, and accelerations of the components involved, taking into account the effects of forces like tension in the pulley and the restoring force of the spring.

How do you set up the equations of motion for a pulley-spring system?

To set up the equations of motion, you start by identifying the forces acting on each component of the system. For a typical pulley-spring system, you would apply Newton's second law (F=ma) to each mass, considering the tension in the pulley and the spring force. You also need to relate the displacements of the masses to the spring's extension or compression using Hooke's law.

What role does the spring constant play in the analysis?

The spring constant, denoted as 'k', is a measure of the stiffness of the spring. It plays a crucial role in determining the force exerted by the spring when it is compressed or extended. In the equations of motion, the spring force is calculated as F_spring = -kx, where 'x' is the displacement from the spring's equilibrium position. A higher spring constant results in a larger restoring force for a given displacement.

How do you determine the equilibrium position of the system?

The equilibrium position of the pulley-spring system is found by setting the net force acting on the system to zero. This involves balancing the forces due to the weights of the masses, the tension in the pulley, and the spring force. By solving the resulting equations, you can find the position where the system remains at rest, which corresponds to the point where the forces are in balance.

What are the common applications of absolute motion analysis in pulley-spring systems?

Absolute motion analysis of pulley-spring systems is commonly applied in various engineering fields, including mechanical design, robotics, and biomechanics. It is used to design systems such as elevators, cranes, and suspension systems, where understanding the dynamics of moving parts is essential for ensuring safety, efficiency, and performance.

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