How do we prove Simple Harmonic Oscillation in a hanging spring?

[EBBAzores]

I have a question that's being bugging me around. This might be simple but I can't figure it out. If there's a spring hanging from the ceiling and we want to prove that there's a Simple Harmonic Oscilation then why don't we account for the gravitational force?

The sum of forces equals to mass times aceleration (ma) according to Newton's 2nd law, so the sum of forces is the gravitational force plus the force by the spring (there should be vectors above the forces of course).

This is a differential equation since aceleration equals the second derivative of position and to prove that there is a S.H.O. the equation must be
(d²x)/(dt²) + (K/m)x = 0

But in fact what I actually get is
(d²x)/(dt²) + (K/m)x + g = 0

I know that in this equation g is a constant but my question is is g relevant if we want to prove that there is a S.H.O?

And following that could anybody tell me how do we demonstrate that the period is in fact
T=2π√(m/K)

Thanks for your time

 

[tiny-tim]

Welcome to PF!

Hi EBBAzores! Welcome to PF!

(have a pi: π and a square-root: √ and try using the X² button just above the Reply box )

(d^2x)/(dt^2) + (K/m)x + g = 0

Hint: put y = x + gm/K

 

[Dale]

Just do a change of variables with x=X-gm/k

tiny-tim got me!

 

[EBBAzores]

Ok one problem down, so i'll assume that g isn't relevant since I change variables for the S.H.O. but now how do we prove that the period is in fact the above one?

P.S.
Thanks for welcoming me in here =)

 

[tiny-tim]

… how do we prove that the period is in fact the above one?

uhh?

solve the equation! ​

 

[EBBAzores]

I've solved it. Now I get the actual S.H.O. equation which is now
(d²y)/(dt²) + (K/m)y = 0

That part I understood but now I want to know why the period of oscillation is T=2π√(m/k)
how do we demonstrate that?

I can't figure it out sorry for wasting your time with this, maybe I'm really tired and that's why I can't figure it out

 

[tiny-tim]

I've solved it. Now I get the actual S.H.O. equation which is now
(d²y)/(dt²) + (K/m)y = 0

ok, so y'' = -(K/m)y …

the solutions are … ? ​

 

[EBBAzores]

This might be a REALLY wild guess but where it goes xD

(d₂y)/(dt₂) = -(K/m)y

so that meand that

dt₂ =(d₂)y/(-k/m)y the y's disappear so I'll assume now that dt is an aproximation to the period T and that the minus sign before k doesn't really interest us so we'll exclude it therefore

T²=(d₂)/(k/m) and now

T=1/√(k/m) and to put it in angular references T=2π/√(k/m)

I don't know if this is right our am I getting dumber xp problaby I need some sleep

 

[tiny-tim]

(you're confusing the X² button with the X₂ button )

(d²y)/(dt²) = -(K/m)y

so that meand that

dt² =(d²)y/(-k/m)y the y's dissapear

no they don't disappear!

you can't possibly do that!

I don't know if this is right our am I getting dumber xp problaby I need some sleep

yes, get some sleep :zzz:

(then think about what solutions of shm you know)

 

[HallsofIvy]

You have been told repeatedly "solve the equation" but have not. Do you not know the general solution to y''= ay?

 

[EBBAzores]

Ok so here I am again ready to finish it already.

Since we know that y=Acos(ωt+$\phi$) so (d²y)/(dt²)=-Aω²cos(ωt+$\phi$) we can easily get that

-Aω²cos(ωt+$\phi$)= -(k/m)(Acos(ωt+$\phi$))

from that expression we obtain that ω²=k/m concluding that ω=√(k/m)

we also know that T=2π/ω=2π√(m/k)

I think that this time its really finished!