- #1
ShakyAsh
- 6
- 0
Okay, so I've been trying to figure out what would be the time period of spring of mass 'M' & spring constant K which is hung vertically from a ceiling.
Dividing the spring into n equal massless springs of spring constant k=nK which are connected to each other by n small objects of mass m=M/n like this-
Δx1=mg/k
Similarly,
Δx2=2mg/k
...
Δxn=n mg/k
ΔX(Total)=ƩΔx=mg n(n+1)/2k
Putting values of m and k
ΔX=Mg n(n+1)/2Kn^2
Taking the limit n→∞
ΔX=Mg/2K which corresponds to an effective mass of M/2 rather than M/3 as given in Wikipedia & other websites.
I would be grateful if someone would be able to explain what i am doing wrong.
Dividing the spring into n equal massless springs of spring constant k=nK which are connected to each other by n small objects of mass m=M/n like this-
_____
n | (k)
m
:
:
3 | (k)
m
2 | (k)
m
1 | (k)
m
Now, extension in first spring n | (k)
m
:
:
3 | (k)
m
2 | (k)
m
1 | (k)
m
Δx1=mg/k
Similarly,
Δx2=2mg/k
...
Δxn=n mg/k
ΔX(Total)=ƩΔx=mg n(n+1)/2k
Putting values of m and k
ΔX=Mg n(n+1)/2Kn^2
Taking the limit n→∞
ΔX=Mg/2K which corresponds to an effective mass of M/2 rather than M/3 as given in Wikipedia & other websites.
I would be grateful if someone would be able to explain what i am doing wrong.