When a spring mass system is connected vertically with two massless springs in series whose spring constants are $k_1$ and $k_2$ to a block of mass $m$ we know that equal forces act on both the springs. Let that force during oscillations be $F$.
When we calculate effective spring constant $k_s$, why don't we say the net force acting on the system is $2F$?
Finding net force acting on the above system:
When the block is attached,the system attains equilibrium position through displacements $x'_1$ and $x'_2$.
At equilibrium: $2F'=mg$(Where $F'$ is magnitude of spring force initially by each spring)
So, $k_1x'_1+k_2x'_2=mg$ (equation 1)
When the system is pulled down it makes oscillations,now:
Total elongation be $x$
Elongation in spring 1 be $x_1$ and elongation in spring 2 be $x_2$.
Total spring force $= -k_1x'_1-k_2x'_2-k_1x_1-k_2x_2$
Total forces acting on the system $= -k_1x'_1-k_2x'_2-k_1x_1-k_2x_2+mg = -mg-k_1x_1-k_2x_2+mg$
(from equation 1)
So, total force $= -k_1x_1-k_2x_2 = F_1+F_2=2F$(as we know that both forces are equal)
So net force acting on the system is $2F$
The way I calculated effective spring constant is:
$2F/k_s = F/k_1 + F/k_2$
$2/k _s = 1/k_1 +1/k_2$
But that is not a correct equation. What's wrong in taking net force acting on system as $2F$.