# What is the spring constant in parallel connection and series connection?

##### 1 Answer

**Parallel**.

When two massless springs following Hooke's Law, are connected via a thin, vertical rod as shown in the figure below, these are said to be connected in parallel. Spring 1 and 2 have spring constants

This system of two parallel springs is equivalent to a single Hookean spring, of spring constant

#k=k_1+k_2#

**Series**.

When same springs are connected as shown in the figure below, these are said to be connected in series. A constant force

This system of two springs in series is equivalent to a single spring, of spring constant

For spring 1, from Hooke's Law

#F=k_1x_1#

where

Similarly if

#F=k_2x_2#

Total deformation of the system

#x_1+x_2=F/k_1+F/k_2#

#=>x_1+x_2=F(1/k_1+1/k_2)#

Rewriting and comparing with Hooke's law we get

#k=(1/k_1+1/k_2)^-1#