Question

Derive the expression for resultant spring constant when two springs having constant k₁ and k₂ are connected in parallel.

Answer 1

k₁ and k₂ 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 mass m 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 F₁ and second spring be elongated by the same displacement x due to force F₂, then the net force

F = – k₁ x – k₂ x …..(2) 

Equating equations (2) and (1), we get k_p = k₁ + k₂ …..(2) 

Generalizing, for n springs connected in parallel,

 If all spring constants are identical i.e., k₁ = kk₂ = … = k_n = k then

This implies that the effective spring constant increases by a factor n. Hence, for the springs in parallel connection, the effective spring constant is greater than individual spring constant.