Equation of Motion for 2 Springs with Different Constants & Mass Attached

[brad sue]

Please , I need to set up the equation for two springs.

The first one is attached to a ceiling and has a constant k. The second one is attached at the tail of the first one and has a spring constant k'.

If a mass m is attached to the second spring, How can I set up the equation for the system?

 

[lightgrav]

the bob will be pulled downward by Earth's gravity.
It will accelerate downward until the Tension in spring #2 is equal to mg .
How far has the bob moved by now? What is the effective "k"?

 

[brad sue]

the bob will be pulled downward by Earth's gravity.
It will accelerate downward until the Tension in spring #2 is equal to mg .
How far has the bob moved by now? What is the effective "k"?

well I see what you mean but I don't know the answer.

I think that the bob will move down by mg=-kx----> x=mg/k''
k'' is the new constant that I don't know ...

please help me because I don't understand how the system reatcs...

B

 

[myxinjie]

Firstly, you should solve the whole system's K .
Briefly, F1 is the first spring's force;
F2 is the second spring's force;
then you get F1 = k*Deta X1
F2 = k'*Deta X2
but if the two springs connected, the force will be the same.
so k*Deta X1 = k'*Deta X2 = F
and Deta X = Deta X1 + Deta X2 = F/k+ F/k' = F(k+k')/kk'
so F=Deta X * kk'/(k+k')
and mX'' = (X-X0) * kk'/(k+k')
and you got it.
good luck