Two Spring System: Steady State Motion

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Homework Help Overview

The problem involves a system of two springs connected by a mass, with a periodic force applied to the upper mass. The inquiry focuses on determining the steady state motion of each mass within the system, characterized by their respective spring constants.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss various methods for analyzing the system, including the use of free body diagrams and the potential equivalence to an electrical circuit. Questions arise regarding the appropriateness of certain equations and whether specific substitutions are valid solutions to the differential equations governing the system.

Discussion Status

Participants are actively engaging with the problem, with some offering clarifications on the relationships between the displacements of the masses and the springs. There is a recognition of the need to ensure that the equations accurately reflect the physical setup.

Contextual Notes

There is mention of the original poster's uncertainty regarding the correct formulation of equations and the interpretation of displacement in relation to the springs and masses. Additionally, the forum encourages the use of specific notation conventions.

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Homework Statement


Consider a system consisting of two springs suspended from the ceiling. The first has a spring constant k-1, the second k-2. They are connected by a mass m and the second spring also has a mass m connected at the bottom. A periodic force is applied to the upper mass. What is the steady state motion for each mass?



Homework Equations


Fcoswt
x1 = Acos wt
x2 = Bcos wt


The Attempt at a Solution

 
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Welcome to PF!

Hi mwkfun! Welcome to PF! :wink:

Show us what you've tried, and where you're stuck, and then we'll know how to help. :smile:
 
I tried two different ways...I attempted to turn the system into an electrical circuit equivalent and I tried with free body diagrams and the following equations:
mX=mg + k1x1 + k2x2 - Fcoswt
mX=mg + Fcoswt - k1x1 - k2x2

(For these two eqns. the X means X double dot indicating acceleration)

mX2 = mg + k2x2 - Fcoswt
mX2 = mg + Fcoswt - k2x2

(again, X2 indicates x double dot...I don't know how to type it the correct way)
I think I should be solving for x1 and x2, but I am not sure if I have the correct equations. The eqns. reflect the free body diagrams I have drawn. Thanks for any help you can give.
 
I am also wondering whether I should use the x1= Acoswt and x2 = Bcoswt as a substitutions or if they are specific solutions to the differential eqns.
 
mwkfun said:
… I tried with free body diagrams and the following equations:
mX=mg + k1x1 + k2x2 - Fcoswt
mX=mg + Fcoswt - k1x1 - k2x2

(For these two eqns. the X means X double dot indicating acceleration)

mX2 = mg + k2x2 - Fcoswt
mX2 = mg + Fcoswt - k2x2

Hi mwkfun! :smile:

(on this forum, it's best to use dashes instead of dots: X1'' :wink:)

I don't understand why you've put each of these equations in pairs :confused:

Anyway, you need to take into account that the displacement of the lower mass is not the same as the displacement of the lower spring … it's the displacement of the lower spring minus the displacement of the upper spring :smile:
 
Thanks so much for your help. I think I got it!
 

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