What is the physical significance of A and B in mass spring SHM solutions?

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    Mass Shm Spring
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Discussion Overview

The discussion revolves around the physical significance of the constants A and B in the solutions for undamped simple harmonic motion (SHM) of a mass-spring system. Participants explore the mathematical forms of the solutions and their implications for understanding the system's behavior at different initial conditions.

Discussion Character

  • Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant, Glenn, questions the physical significance of A and B in the solutions x = Acos(wt) and x = Bsin(wt), noting that A is typically seen as the amplitude depending on the initial conditions.
  • Another participant points out that the second term in the solution should be sin() instead of cos(), which prompts a clarification from Glenn.
  • Glenn later suggests that if the initial position is zero after a quarter cycle, then B must equal zero, simplifying the solution to x = Acos(wt) and confirming that C = A.
  • Another participant comments that the constants A and B represent boundary conditions, such as the initial position and velocity of the oscillator, emphasizing their role in the general solution of the second-order differential equation.
  • One participant mentions that the constants can also be expressed in terms of amplitude and phase information, indicating a broader interpretation of the solutions.

Areas of Agreement / Disagreement

Participants express differing views on the interpretation of A and B, with some focusing on their roles in initial conditions and others discussing their representation of amplitude and phase. The discussion remains unresolved regarding a singular interpretation of these constants.

Contextual Notes

The discussion highlights the dependence on initial conditions and the mathematical representation of SHM, but does not resolve the implications of these constants in a definitive manner.

Glenn G
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Hi community,
I've been looking at solutions for mass spring shm (undamped for now) ie that

x = Acoswt and x = Bcoswt work as solutions for dx2/dt2 = -(k/m)x

and that the general solution is the sum of these that with a trig identity can be written as

x = C Cos(wt - φ) where C is essentially the amplitude (and is given by √(A2 + B2)

My question is the physical significance of A and B in the two separate solutions above (before this youtube video) I've always gone for the solutions as either the Acoswt or Asinwt (with A being the amplitude) depending on where the mass is in its oscillating cycle at time t=0, i.e. would have gone with the coswt one if

x = +A at t=0.

If I let A and B both be A then my factor C (amplitude) comes out as √(2)A where I want it to represent the Amplitude A.

Would really appreciate help.
regards,
Glenn.
 
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Glenn G said:
(before this youtube video)
Doesn't look like the video link is working...?
 
Glenn G said:
x = Acoswt and x = Bcoswt work as solutions
and is the 2nd term supposed to be sin()?
 
berkeman said:
and is the 2nd term supposed to be sin()?
Sorry yes I did mean Bsinwt for the second solution...

In fact I think it is fine, because if you state that x = 0 after quarter of a cycle (so ∏/2 radians) then you get 0 = Bx1 therefore B = 0 (in x = Acoswt + Bsinwt) so the sine term disappears from the general solution anyway and you end up with C = √A^2 = A for the coefficient which is what I wanted.

thanks,
G.
 
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Yeah, it just means that the spring doesn't necessarily have zero phase angle at t=0. It's a general way to express a sinusoid that has amplitude and phase information... :smile:
 
Note that because we start out with a 2nd order differential equation the general solution will contain two constants. They can be A and B, or A and ##\phi##, or whatever. Their physical interpretation is that they set the boundary conditions, for example the initial position and velocity of the oscillator.
 
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