Effective mass (spring–mass system)
In a real spring–mass system, the spring has a non-negligible mass formula_1. Since not all of the spring's length moves at the same velocity formula_2 as the suspended mass formula_3, its kinetic energy is not equal to formula_4. As such, formula_1 cannot be simply added to formula_3 to determine the frequency of oscillation, and the effective mass of the spring is defined as the mass that needs to be added to formula_3 to correctly predict the behavior of the system. Ideal uniform spring. The effective mass of the spring in a  spring-mass system when using an ideal spring of uniform linear density is 1/3 of the mass of the spring and is independent of the direction of the spring-mass system (i.e., horizontal, vertical, and oblique systems all have the same effective mass). This is because external acceleration does not affect the period of motion around the equilibrium point. The effective mass of the spring can be determined by finding its kinetic energy. This requires adding all the mass elements' kinetic energy, and requires the following integral, where formula_8 is the velocity of mass element: Since the spring is uniform , formula_10, where formula_11 is the length of the spring at the time of measuring the speed. Hence, The velocity of each mass element of the spring is directly proportional to length from the position where it is attached (if near to the block then more velocity and if near to the ceiling then less velocity), i.e. formula_14, from which it follows: Comparing to the expected original kinetic energy formula formula_19 the effective mass of spring in this case is "m"/3. Using this result, the total energy of system can be written in terms of the displacement formula_20 from the spring 's unstretched position (ignoring constant potential terms and taking the upwards direction as positive): Note that formula_23 here is the acceleration of gravity along the spring. By differentiation of the equation with respect to time, the equation of motion is: The equilibrium point formula_25 can be found by letting the acceleration be zero: Defining formula_27, the equation of motion becomes: This is the equation for a simple harmonic oscillator with period: So the effective mass of the spring added to the mass of the load gives us the "effective total mass" of the system that must be used in the  standard formula formula_30 to determine the period of oscillation. General case. As seen above, the effective mass of a spring does not depend upon "external" factors such as the acceleration of gravity along it. In fact, for a non-uniform spring, the effective mass solely depends on its linear density formula_31 along its length: So the effective mass of a spring is: This result also shows that formula_37, with formula_38 occurring in the case of an unphysical spring whose mass is located purely at the end farthest from the support. Real spring. The above calculations assume that the stiffness coefficient  of the spring does not depend on its length. However, this is not the case for real springs. For small values of formula_39, the displacement is not so large as to cause elastic deformation. Jun-ichi Ueda and Yoshiro Sadamoto have found that as formula_39 increases beyond 7, the effective mass of a spring in a vertical spring-mass system becomes smaller than Rayleigh's value formula_41 and eventually reaches negative values. This unexpected behavior of the effective mass can be explained in terms of the elastic after-effect (which is the spring's not returning to its original length  after the load is removed).