Mass Affect the Period of a Spring

As we all know that springs find their application in almost all types of machines, they can be fitted in two ways:

1. In Series
2. In Parallel

Before going into the depth of the arrangement and the combination of springs, let us first understand what spring is and why is it important?

Spring- Composition and usage

In layman’s language, we all say that the metal spiral we see every day is spring, but in the actual term, it is different.

In the language of physics, spring is a kind of medium that is capable of returning to its equilibrium position when it is displaced from its original position. 

In other words, it returns to the equilibrium point if one touches it and moves it slightly to a new distance.

From this concept, it is clear that the assumption we have of spring is far away from the real meaning of the springs.

Now the question is, how does a spring go back to the equilibrium position?

Does it have any driving force that drives it back to the equilibrium position?

Yes. A spring possesses a force of its own that is known as the restoring force.

The restoring force present in the spring is its force that is possessed by the spring by virtue.

It is because of this force only that the spring tends to regain its shape and original position.

The restoring force is largely dependent on the magnitude of the displacement. The greater the displacement, the greater the restoring force applied by the spring to attain its equilibrium position. It is to be noted that the spring applies this force in the direction that is opposite to the displacement.

The standard formula derived for the restoring force is:

F= -kx

where F denotes the restoring force

k is the spring constant that depends on the material of the spring

x is the displacement caused in the spring 

Here the negative sign denotes that the spring applies the restoring force in the opposite direction to the displacement.

This formula, commonly known as Hooke’s law, was given by Robert Hooke.

There are basically four types of springs present :

1. Compression
2. Extension
3. Torsion
4. Constant force springs

Just like the four types of springs, there are two different ways of combining a spring.

Springs in series and parallel

There are two possible arrangements of springs :

1. Two or more springs can be combined in a series combination
2. Two or more springs can be combined in a parallel combination

Series Combination – When two massless springs are combined in a series that is joined one after the other, and if a constant force F is applied on the second spring, then both the springs get extended. 

The total extension observed is the sum of the elongation caused by force in each spring. 

By using Hooke’s law for the spring system, we know that F= kx

So, for spring 1, we have F= k1 x1

x1 is the deformation caused in the first spring

And for spring 2, we have F= k2 x2

Here also x2= deformation caused in the second spring

The spring constant is found to be different for both springs. 

So the formula derived for the effective spring constant in the case of series combination in springs is :

1/k = ( 1/k1 + 1/k2) 

And the formula for total deformation is :  x1+ x2= F( 1/k1 + 1/k2) 

Parallel Combination

When the two massless springs are not connected one after the other, but for instance, one is connected above and the second is connected below, both are joined together by a thin rod vertically. Such a type of connection is known as Parallel Combination. 

It is to be noted here that the spring constants for both the springs are different, that is, k1 and k2, respectively.

Now when a force F is applied on a thin rod that joins both the rods, it is obvious for both the springs to stretch by the same amount. The force F that is applied on the rod is perpendicular to the rod. 

In this kind of a system, the two springs are considered as a single spring, and the formula for the Parallel spring constant (effective spring constant) is the simple sum of both the spring constants, i.e. k= k1+ k2

It is interesting to note that the restoring force is different on each spring in a parallel combination of springs, but the displacement caused is the same in each spring for a combination of springs in parallel.

Conclusion

Both the parallel as well as the series combination of springs follow Hooke’s law and have a different value of effective spring constant and restoring force. 

In both types of combination, it is all about the alignment of the springs that determine whether they’ll work in series or parallel to each other.