Time period of a heavy spring with an attached mass at the end

[thephysicist]

I want to know the way to derive the time period equation of a spring mass system accounting for the mass of the spring but not using the energy analysis method but by proceeding in the same way as we do by ignoring the mass of the spring. Please help. I did not find any texts at my level. Any links would suffice gratefully.

 

[Nugatory]

What assumptions are you making about the distribution of mass along the length of the spring?

 

[thephysicist]

Uniformly distributed.

 

[sophiecentaur]

I want to know the way to derive the time period equation of a spring mass system accounting for the mass of the spring but not using the energy analysis method but by proceeding in the same way as we do by ignoring the mass of the spring. Please help. I did not find any texts at my level. Any links would suffice gratefully.

I did a Google search on "mass on heavy spring" and found loads of hits. I am not sure what your level is but how does this link suit you (it was top of my list)?

 

[thephysicist]

Thanks for the link but this derivation also uses energy analysis. I want to find an approach that is similar to deriving the equation of motion of a massless spring with an attached mass.

 

[AlephZero]

The equations of motion for the spring without an extra mass on the end are basically the same derivation as the wave equations for longitudinal waves (not transverse waves in a string under tension).

The mass on the end just changes the boundary conditions at the end.

You will find plenty of references to an approximate solution that includes m/3, where m is the mass of the spring. The approximations is to assumes the spring mass is small enough not to affect the vibrating shape of the spring. That seems to be what sophiecentaur's reference is doing, but it just states the m/3 factor without attempting to justify it.