Spring Mass System

Spring-mass systems

In physics, when a block or an object is hung or suspended from a spring, a lot of equations and derivations have arrived from the same. The spring-mass system examples are based on this premise. When discussing simple harmonic motion in physics, the restoring force working on the moving object is directly proportional to its magnitude of displacement. It works towards the moving object’s equilibrium position. In simpler terms, this will result in the oscillatory movement of the object, which is not affected by friction or any other energy and continues indefinitely. 

While understanding spring-mass system examples and solving spring-mass system derivations, let us first understand the concept of effective mass. Now, in a real spring-mass system, the spring will have a non-negotiable mass. Let us assume m. We know that not the entirety of the spring length will move at the same velocity (v) as that of the suspended mass (M), the kinetic energy for the same cannot be ½ mv square. Thus, effective mass is the mass of the spring is the mass that is required to be added to M in order to predict the system’s behaviour in a correct way. 

Vertical and Horizontal Spring Mass System

When talking about spring-mass system examples and spring-mass system equations, we have the vertical and horizontal spring-mass systems. 

Now, it is essential to know that when there is no friction present, and if the masses and springs are the same, both the systems will oscillate identically around an equilibrium position. Although, when it comes to vertical spring, we need to add the factor of gravity as it can compress or even stretch the spring beyond its usual length to the equilibrium position. And thus, when we are able to find this displaced position, we can set the displacement as y=0 and treat the vertical spring similar to the horizontal spring.  

Horizontal oscillation of spring

Let us assume a mass (m) is attached to one end of the spiral spring while the other end of the spring is fixed to a support(L). The body is kept on a horizontal surface that is smooth. Let the body be displaced through a given distance x towards the (R) side and released. The body will oscillate in its mean position. There is going to be a restoring force acting in the opposite direction, which will be proportional to the displacement.

In order to analyse the vertical motion of a spring-mass system, let the force on the spring be represented as Fs  = kx and gravitational force as Fg  = mg.

Using Newton’s second law, 

The net force on the block = 

ΣF = ma

=  Fs – Fg

= kx-mg

As the block or weight does not accelerate, kx-mg= 0

Conclusion

To conclude, spring-mass system examples are important in physics and mathematics as spring-mass system equations are derived, be it the vertical spring-mass system, the horizontal spring-mass system and springs in series and parallel combination. When there is no friction present, considering the spring and masses are the same, both the horizontal and vertical spring-mass system will oscillate around an equilibrium position in the same way.