Derive the expression for resultant spring constant when two springs having constant `k_(1) and k_(2)` are connected in parallel.

Explanation

`k_(1) and k_(2)` attached to a mass m as shown in figure. The results can be generalized to any number of springs in parallel.

Let the force F be applied towards right as shown in figure. In this case, both the springs elongate or compress by the same amount of displacement. Therefore, net force for the displacement of massm is
`F=-k_(P),x" "...(1)`
where `k_(P)` is called effective spring constant.
Let the first spring be elongated by a displacement x due to force `F_(1)` and second spring be elongated by the same displacement x due to force `F_(2)`, then the net force
`F=-k_(1)x-k_(2)x" "...(2)`
Equating equations (2) and (1), we get
`k_(P)=k_(1)+k_(2)" "...(3)`
Generalizing, for n springs connected in parallel,
`k_(P)=underset(i=1)overset(n)Sigmak_(i)" "...(4)`
If all spring constants are identical i.e., `k_(1)=k_(2)=...=k_(n)=k` then
`k_(P)=nk" "...(5)`
This implies that the effective spring constant increases by a factor n. Hence, for the springs in paralle connection, the effective spring constant is greater than individual spring constant.