Question

What kind of combination of springs does this arrangement in figure shown belong to

A. Series
B. Parallel
C. Hybrid (series and parallel)
D. None of the above

Answer 1

We could find the combination represented by the given system from the equivalent spring constant of the system. For that, you could consider elongating the upper spring by some distance y and also the resultant displacement in the lower spring. You could find the restoring force in each case and thereby the net restoring force of the system.

Formula used:
Hooke’s law,
$F=-ky$

Complete solution:
In the question, we are given a spring-mass system that contains a mass and two springs which is hung vertically. We are supposed to find the combination of springs that the given arrangement is representing. For that, let us consider that the upper spring of spring constant ${{K}_{1}}$ is being pulled downwards and is undergoing a displacement of y. The restoring force in this upper spring is given by Hooke’s law, that is,
${{F}_{1}}=-{{k}_{1}}y$

Due to spring constrained motion, as a result of this displacement of the upper spring, the lower spring of spring constant ${{K}_{2}}$ is being compressed by a distance y. So the restoring force in the lower spring can also be given by Hooke’s law as,
${{F}_{2}}=-{{k}_{2}}y$

As the upper spring is being displaced downwards, the restoring force will be directed upwards as the spring is being elongated. Now for the other spring that is being compressed, the restoring force will be again directed upwards. So, the net restoring force of the system is given by the sum of the restoring forces of the two springs.
${{F}_{net}}={{F}_{1}}+{{F}_{2}}$
$\Rightarrow {{F}_{net}}=-{{k}_{1}}y-{{k}_{2}}y$
$\therefore {{F}_{net}}=-\left( {{k}_{1}}+{{k}_{2}} \right)y$

So, we see that the equivalent spring constant of the system is the sum of the spring constants of the two springs. That is,
${{k}_{eq}}={{k}_{1}}+{{k}_{2}}$

This would have been the equivalent spring constant if the springs had been connected parallel to each other. Hence, we found that the given system represents a parallel combination of springs.

Note:
The Hooke’s law states that the restoring force developed as the result of the spring being stretched or compressed is directly proportional to the resultant displacement. The spring constant (k) is the constant of proportionality introduced in this relation. The spring constant of a spring gives measure of stiffness of the spring.