So, this is a post regarding a general result which is of a lot of help if remembered in Mechanics problems. The derivation is of how to calculative effective mass of block and spring attached if spring is massive.
Let's consider the mass of spring to be .
So let's begin,
So let us take a small element of mass of the spring, so total energy can be calculated by adding all the length elements' kinetic energy, and requires the following integral:
Since spring is uniform,
The velocity of each mass element of the spring is directly proportional to its length, i.e.
Comparing to the expected original kinetic energy formula,
we can conclude that effective mass of spring in this case is
The EFFECTIVE MASS of the spring in a spring-mass system when using an ideal spring of uniform linear density is of the mass of the spring.
And we're done.