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

AI Thread Summary
The discussion focuses on deriving the time period equation for a spring-mass system while accounting for the spring's mass, specifically avoiding energy analysis methods. The user seeks a derivation method similar to that used for massless springs, emphasizing the need for clarity on assumptions regarding mass distribution along the spring. References suggest that the mass of the spring can be approximated with a factor of m/3, where m is the spring's mass, under the assumption that the spring's mass does not significantly alter its vibrating shape. The conversation highlights the challenge of finding suitable resources that meet the user's level of understanding. Overall, the thread underscores the complexity of including spring mass in dynamic equations without relying on energy methods.
thephysicist
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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.
 
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What assumptions are you making about the distribution of mass along the length of the spring?
 
Uniformly distributed.
 
thephysicist said:
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)?
 
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.
 
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.
 

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