Hooke's Law: How to find Amplitude

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Hooke's Law relates to the motion of springs, described by the differential equation x'' + wx = 0. The general solution is A sin(wt) + B cos(wt), where A and B are constants representing the initial conditions of the system. The amplitude of the motion is calculated as the square root of the sum of the squares of A and B, or sqrt(A^2 + B^2). This relationship arises from the trigonometric identity used to express the solution in a more recognizable form, A cos(ωt + φ). Understanding these constants is essential for grasping the physical meaning of amplitude in simple harmonic motion.
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Hello all,
(Newbie here so I hope I'm posting this in the right section, if not please guide me to the correct one)

I'm studying Hooke's Law in Physics and there is something I can't understand.
The equation for the Simple Harmonic Motion of a spring is given by x'' + wx =0

the general solution for the Differential equation above is given by: A sin (wt) + B cos (wt)

I came across a reference book that says: the amplitude is given by:
sqrt {A^2 + B^2}

I'd like to understand how they got this?? Right now I know that w is the angular frequency and wt gives the phase angle, but A and B as constants make no sense to me, what are they and what do they mean physically?

Any kind of answer would be greatly appreciated, please help me understand!
Thanks in advance!
 
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A more recognizable form of the general solution would be this:
A\cos (\omega t + \phi)
Where A is the amplitude. Expand this (using a trig identity) and compare with the form you were given.
 
Thanks a lot, I got it now :D
 
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