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SHM of Mass Oscillating between Two Identical Springs

  • #1
1. The problem statement, all variables and given/known
A mass m is connected between two identical springs (along a y-axis) with identical spring constants k. The equilibrium length of each spring is L, but they are stretched to twice this length when m is in equilibrium. By analyzing the force acting on the mass when it is displaced by a small distance x, find the equation of motion of the mass and thus find the angular frequency of oscillation. Ignore gravity.

Hint: You will have to make an approximation, but you only need to keep the term that is linear in x, higher powers of x should be ignored.


Homework Equations


F_sp = -kx = ma


The Attempt at a Solution


When you displace the mass a small x from equilibrium, the y components of the spring forces cancel, leaving twice the spring force in the x direction.

If the angle made from the equilibrium (horizontal) to the stretched state is [tex]\theta[/tex] and the displacement is positive x, then

- 2Fsp sin[tex]\theta[/tex] = max

I'm really confused about what exactly the spring force is: do I need to consider that the spring is stretched or is that irrelevant? And how does the approximation apply to x (aren't we approximating [tex]\theta[/tex]?). Can someone lead me in the right direction?

EDIT:
I found the magnitude of the spring force to be k(2L-L) = kL.
I approximated sin[tex]\theta[/tex] to be [tex]\theta[/tex] = x/2L
So then the net force in the x direction on the mass = -2Fspsin[tex]\theta[/tex] = -kx = max
So then the equation of motion is just the differential equation for SHM for a mass on a spring?

Is this correct?
 
Last edited:

Answers and Replies

  • #2
ehild
Homework Helper
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Sketch the arrangement, please.

ehild
 
  • #3
http://webserv.kmitl.ac.th/~physics/mb/images/stories/publisher/witoon/Problem_oscillation.pdf [Broken]

Essentially question #53, except with springs.
 
Last edited by a moderator:
  • #4
ehild
Homework Helper
15,477
1,854
Your solution is all right.

ehild
 
  • #5
Great, thanks!
 

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