SHM of Mass Oscillating between Two Identical Springs

In summary, the problem statement is that a mass is connected between two springs and when it is displaced by a small distance, the y components of the spring forces cancel and the springs are stretched. When analyzing the force acting on the mass when it is displaced by a small distance, they found that the equation of motion is the differential equation for SHM for a mass on a spring.
  • #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:
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  • #2
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.
 
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  • #4
Your solution is all right.

ehild
 
  • #5
Great, thanks!
 

What is SHM?

SHM stands for Simple Harmonic Motion, which is a type of periodic motion where an object oscillates back and forth around a central equilibrium position.

What is the equation for SHM?

The equation for SHM of an object oscillating between two identical springs is x(t) = Asin(ωt + φ), where x is the displacement of the object, A is the amplitude, ω is the angular frequency, and φ is the phase angle.

What is the significance of the angular frequency in SHM?

The angular frequency, ω, is a measure of how fast the object is oscillating. It is directly proportional to the frequency of oscillation and inversely proportional to the period of oscillation.

How does changing the mass affect SHM?

In SHM, changing the mass of the object does not affect the angular frequency or the period of oscillation. However, it does affect the amplitude of oscillation, with a larger mass resulting in a smaller amplitude and vice versa.

What is the relationship between SHM and Hooke's Law?

SHM is closely related to Hooke's Law, which states that the force exerted by a spring is directly proportional to its displacement from equilibrium. In SHM, this force is what causes the oscillation of the object between the two springs.

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