Basic Spring/Oscillator Question

  • Thread starter wripples
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In summary, the problem given involves a mass connected to two springs with different spring constants and equilibrium lengths. The right spring's constant is instantaneously changed and the goal is to find the resulting function for the motion of the mass. One approach to solving this problem is to calculate everything relative to the new equilibrium position and use the initial conditions to find the amplitude and frequency. Solving the resulting differential equation can be a challenging task.
  • #1
wripples
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Homework Statement



So the problem is that you have a mass m connected to two springs, where each of the two springs is connected to a wall, as such:

|--------M--------|

Each spring first has a spring constant k and a equilibrium length L, and then when set up in the above configuration, M is 2L away from both walls (so both springs are stretched a length L from their equilibrium). Then the spring on the right has its spring constant instantaneously changed to 3k (instead of k), and the goal is to find the resulting function to describe the motion of M. We are to take its initial position to be 0 (so x(0) = 0).

2. The attempt at a solution

So my approach is to first set up the problem. First, take positive x to be displacement to the right. We have x(0) = 0, and since the right spring constant is supposed to instantaneously change to 3k, I assumed v(0) = 0. On the other hand, I'm not entirely sure that is right; perhaps I should solve for the acceleration at time t = 0 using the fact that both springs are stretched L past their equilibrium?

Afterwards, I try to find the net force. On the left, the spring with spring constant k has restoring force F_1 = -k(L+x), since it is already displaced by L to begin with, so it should already be acting with a force of -kL when we are at time t = 0, and stretching to the right (positive x) should increase the restoring force. Similarly, the force of the spring on the right is F_2 = -3k(-L+x); at the start, it should already have a force of magnitude 3kL pointing to the right, so it makes sense for x = 0, and as x increases (moves right), its restoring force should decrease.

Now, equating with F_2 - F_1 = F_net = ma, we have ma = 2kL - 4kx, which seems like its a particularly nasty differential equation to solve, due to the inhomogeneous term 2kL. Perhaps this is correct? And if so, how does one such solve a differential equation? I'm particularly rusty on doing that type of stuff.

Thanks!
 
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  • #2
I suggest to calculate everything relative to the new equilibrium position.
You can find this position follows from F=0 (or a=0)
Then you know the solution for motion around the equilibrium. The initial position is L/2 from equilibrium (if your formula for acceleration is right) so the amplitude will be L/2. Find the frequency from the new net force. Use initial conditions to find initial phase.
Then you can add L/2 in order to shift back the origin in the middle.
 
  • #3


Dear student,

Thank you for your question. It seems like you have a good understanding of the problem and are on the right track with your approach. Your initial conditions for the position and velocity of the mass are correct, and you are also correct in considering the acceleration at time t = 0.

As for solving the differential equation, there are a few methods you can use. One common approach is to use the method of undetermined coefficients, where you assume a solution of the form x(t) = A cos(ωt) + B sin(ωt) and then solve for the coefficients A and B. Another method is to use the method of variation of parameters, where you assume a solution of the form x(t) = y(t) cos(ωt) + z(t) sin(ωt) and then solve for the functions y(t) and z(t).

I suggest looking up examples of solving second-order differential equations with inhomogeneous terms using these methods to get a better understanding of the process. It may also be helpful to consult with your instructor or a tutor for further guidance.

Best of luck with your homework!

Sincerely,
 

1. What is a spring oscillator?

A spring oscillator is a physical system that consists of a mass attached to a spring, allowing it to oscillate back and forth around an equilibrium point.

2. What is the equation for the motion of a spring oscillator?

The equation for the motion of a spring oscillator is x = A cos(ωt + φ), where x is the displacement of the mass, A is the amplitude, ω is the angular frequency, and φ is the initial phase angle.

3. How does the stiffness of a spring affect the motion of a spring oscillator?

The stiffness of a spring, also known as its spring constant, affects the motion of a spring oscillator by determining the frequency of oscillation. A stiffer spring will result in a higher frequency and a shorter period, while a more flexible spring will have a lower frequency and a longer period.

4. What factors can affect the period of a spring oscillator?

The period of a spring oscillator can be affected by several factors, including the mass of the object attached to the spring, the stiffness of the spring, and the amplitude of the oscillations. The period is also affected by external forces such as friction and air resistance.

5. How is energy conserved in a spring oscillator?

In a spring oscillator, energy is conserved through the exchange between potential energy stored in the spring and kinetic energy of the oscillating mass. As the mass moves towards the equilibrium point, potential energy is converted into kinetic energy, and as it moves away, kinetic energy is converted back into potential energy. This back-and-forth exchange of energy results in a constant total energy throughout the oscillation.

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