- #1
Silimay
In a block-spring system with a block of mass m and a spring of spring constant k, prove that the angular velocity ω of the block = (k/m)^(1/2).
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 canceled 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 :-)
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 canceled 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 :-)