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Effective mass (spring-mass system)
Encyclopedia
In a spring-mass system the mass of the spring 
, as well as the suspended mass 
, have an influence on the motion. However, since not all of the spring moves at the same velocity as the suspended mass, the mass of the spring cannot be simply added to the suspended mass. The effective mass of the spring is the mass which must be added to the suspended mass to correctly predict the behavior of the system.
The effective mass of the spring in a spring-mass system when using an ideal spring is independent whether the direction of the spring-mass system is horizontal, vertical or oblique, is also 1/3 of the mass of the spring, i.e., m/3. This can be shown by integration
:
Its length is
, mass is 
, velocity is 
and so
where
is the length of the spring.
Hence the total kinetic energy
of the spring and mass is
But the velocity of each position of the spring is directly proportional to its length and so
Hence
With comparison of the original kinetic energy
formula
we can conclude that effective mass of spring in this case is m/3, i.e., 1/3 of the mass of the spring.
Since Jun-ichi Ueda and Yoshiro Sadamoto have done the experiment. It is found that, as the ratio 
increases beyond 7, the effective mass of a spring in a vertical spring-mass system becomes smaller than Rayleigh's value 
and eventually reaches negative values. This unexpected behavior of the effective mass can be explainable in terms of the elastic after-effect.
, as well as the suspended mass , have an influence on the motion. However, since not all of the spring moves at the same velocity as the suspended mass, the mass of the spring cannot be simply added to the suspended mass. The effective mass of the spring is the mass which must be added to the suspended mass to correctly predict the behavior of the system.Ideal spring
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
:
Its length is
, mass is , velocity is and sowhere
is the length of the spring.Hence the total kinetic energy
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...
of the spring and mass is
But the velocity of each position of the spring is directly proportional to its length and so
Hence
With comparison of the original kinetic energy
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...
formula
we can conclude that effective mass of spring in this case is m/3, i.e., 1/3 of the mass of the spring.Real spring
However, the above calculations are only suitable for small values of Since Jun-ichi Ueda and Yoshiro Sadamoto have done the experiment. It is found that, as the ratio increases beyond 7, the effective mass of a spring in a vertical spring-mass system becomes smaller than Rayleigh's value and eventually reaches negative values. This unexpected behavior of the effective mass can be explainable in terms of the elastic after-effect.External links
- http://tw.knowledge.yahoo.com/question/question?qid=1405121418180
- http://tw.knowledge.yahoo.com/question/question?qid=1509031308350
- http://hk.knowledge.yahoo.com/question/article?qid=6908120700201
- http://www.goiit.com/posts/list/mechanics-effective-mass-of-spring-40942.htm
- http://www.juen.ac.jp/scien/sadamoto_base/spring.html
- "The Effective Mass of an Oscillating Spring" Amer. J. Phys., 38, 98 (1970)
- "Effective Mass of an Oscillating Spring" The Physics Teacher, 45, 100 (2007)