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

[Glenn G]

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.

 

[berkeman]

(before this youtube video)

Doesn't look like the video link is working...?

 

[berkeman]

x = Acoswt and x = Bcoswt work as solutions

and is the 2nd term supposed to be sin()?

 

[Glenn G]

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.

 

[berkeman]

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...

 

[Herman Trivilino]

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.