- #1

I can prove this easily in the following manner:

F = ma Newton's law

F = -kx Hooke's law

a = -ω^2x

ma = -kx = -mω^2x

k = mω^2

ω^2 = (k/m)

ω = (k/m)^(1/2)

But when I try to prove it using calculus (as my teacher instructed me to do) something goes wrong:

F = ma

F = -kx

X = Acos(ωt + φ) SHM

F = ma = -kx = -kAcos(ωt + φ)

I integrated to get:

mv = -kωAsin(ωt + φ)

Did I do something wrong here? I kept integrating (so that there was a ω^2 term on the right side) and substitued for x = Acos(ωt + φ) and cancelled out x; but then I got ω = (m/k)^1/2. I don't understand why---is there a flaw in the math somewhere? I think I can probably do the proof by simply integrating the SHM equation and substituting it for acceleration, but I'd like to know what I did wrong above.

Thanks for any help :-)