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

1 Answer
Dec 15, 2016

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 k_1k1 and k_2k2 respectively. A constant force vecFF is exerted on the rod so that remains perpendicular to the direction of the force. So that the springs are extended by the same amount. Alternatively, the direction of force could be reversed so that the springs are compressed.

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This system of two parallel springs is equivalent to a single Hookean spring, of spring constant kk. The value of kk can be found from the formula that applies to capacitors connected in parallel in an electrical circuit.

k=k_1+k_2k=k1+k2

Series.

When same springs are connected as shown in the figure below, these are said to be connected in series. A constant force vecFF is applied on spring 2. So that the springs are extended and the total extension of the combination is the sum of elongation of each spring. Alternatively, the direction of force could be reversed so that the springs are compressed.

![notendur.hi.is

This system of two springs in series is equivalent to a single spring, of spring constant kk. The value of kk can be found from the formula that applies to capacitors connected in series in an electrical circuit.

For spring 1, from Hooke's Law

F=k_1x_1F=k1x1

where x_1x1 is the deformation of spring.

Similarly if x_2x2 is the deformation of spring 2 we have

F=k_2x_2F=k2x2

Total deformation of the system

x_1+x_2=F/k_1+F/k_2x1+x2=Fk1+Fk2
=>x_1+x_2=F(1/k_1+1/k_2)x1+x2=F(1k1+1k2)

Rewriting and comparing with Hooke's law we get

k=(1/k_1+1/k_2)^-1k=(1k1+1k2)1