Theoretical Study of the Energies of the Oscillating System with a Well- Distributed Mass of the Spring

The energy of a spring with a well-distributed mass ms is theoretically studied in this paper. The solution of the wave equation is derived in detail, and then the kinetic energy and potential energy of the spring are studied with the wave equation, as well as the kinetic energy of the oscillating mass M. The kinetic energy and potential energy of the spring, and total energy are numerically simulated for different ratios ms/M with considering the spring’s mass, which makes the property of energy of the oscillating system understood easily.


Introduction
Conventional textbook such as college physics, the mass of a spring ms is usually ignored in the study of the oscillation, which leads to an ideal model for describing the spring oscillator. For the practical considerations, however, the spring has mass, which is not negligible with respect to the mass M suspended at its end. The topics on an oscillating period and total energy of oscillating system have been studied for many years [1][2][3][4][5]. Worsnop et al have suggested that ms/3 of the spring mass should be added to the mass M to calculate the oscillation period [6]. Galloni et al have reported that one may suppose the spring mass is null in which case a mass equal to ms/3 must be added at its end to get the same energy [7,8]. However, the kinetic energy and potential energy of a spring in an oscillation system have not been investigated so far when the mass of a spring is taken into account. The aim of this paper is to analyze the influence of the mass of the spring on the simple harmonic motion of the spring mass system. Specifically, we derive a theoretical description of the kinetic and potential energies, and gain an insight into it with different spring-object mass ratios ms/M.
The rest part of this paper is organized as follows. In the section II, we briefly describe the solution of the wave equation. The energy of an oscillation system is studied in detail in the Section III. Conclusions are given in the Section IV.

The Wave Equation and Dynamic Solution
As illustrated in Fig. 1, we consider a spring-object model, which consists of a spring with a welldistributed mass ms and an object with the mass M at the right end of a spring. The length of the spring is L and the elastic constant is k. The object attached to at the right end of the spring is released from its equilibrium position. According to the Hooke's law and Newton's second law; we can get a wave equation.
Second, both law Hooke's law and Newton's second are applied at the right end of the spring (x = L). Then we can obtain an equation , where the minus sign denotes leftward direction of the elastic force. Substituting Eq. (1) into it, we can obtain the other Third, as the system is stationary when t=0, in other words, the velocity of spring is equal to zero when t=0, therefore In addition, the spring is stretched and the deformation of the spring is 0 l when t=0, thus the deformation of the spring at x from the left fixed end of the spring can be expressed as Based on the above information, the boundary and initial conditions of the wave Eq. (1) are The wave Eq. (1) can be solved using the method of separation of variables. We define , and then substitute it into the wave Eq. (1), we can obtain the following equations where 2 ω is the separation constant. Taking the linear combination of them, the general solution of Eq.
(2) can be written as where 0

The Energy of an Oscillating System
Now we study the energy of an oscillating system, including kinetic energy and elastic potential energy of a spring, as well as the kinetic energy of an oscillating mass M [12]. For the kinetic energy of a x L  In addition, potential energy of a spring of each element x ∆ is also studied, each element x ∆ has a potential energy x L Eq. (16) means that the total kinetic energy of a spring Ekm is varied with the time, which can be seen in Fig. 5.
As for total potential energy of a spring, each element dx has an energy     Eq. (19) means the total energy of an oscillating system is constant. In order to understand the energy of oscillating system fully, the kinetic energy of a spring (Ekm) and potential energy of a spring Ep, kinetic energy of mass M (EkM) and the total kinetic energy (E) are plotted with time for different ratios ms/M in Fig. 5. The parameters of the elastic constant k and spring's mass ms are the same as the above model, i.e., the elastic constant is k = 3.0 N/m and the mass of a spring is ms = 0.03 kg. Fig. 5(A) shows that the kinetic energy of the oscillator EkM and the elastic potential energy Ep versus time without considering the mass of the spring (ms = 0) [13]. We can find that kinetic energy EkM and potential energy Ep can transform each other, whereas the total energy of the system E is conserved. That is, the elastic potential energy of the spring Ep becomes zero while the kinetic energy of mass EkM reaches its maximum value. Fig. 5(B) shows the energy distribution of the spring system with the mass M = 0.3 kg. When the elastic potential of the spring Ep reaches its maximum and equals to the total energy E, the kinetic energy of the oscillator EkM and the spring Ekm are both at their minimum, i.e., zero. In addition, the time-varying characteristic of the spring's kinetic energy Ekm is the same as that of the oscillator EkM. That is, they reach their maxima/minima at the same time. Although the kinetic energy of the spring Ekm is much smaller than the kinetic energy of the oscillator EkM, the total energy of the system E is still conserved. The elastic potential energy Ep changes in Figs. 5(C) and 5(D) are the same as that of Figs. 5(A) and 5(B). The kinetic energies of the spring Ekm and the oscillator EkM reach their minimum at the same time. However, the kinetic energy of the oscillator EkM gets its maximum before the spring Ekm. From Figs. 5(E) and 5(F), we can find that the kinetic energies of the oscillator EkM and the spring Ekm increase periodically as the mass of the oscillator M decreases (or increases the ratio ms/M), and the maximum of the spring kinetic energy Ekm is greater than that of the oscillator EkM, whereas the total energy of the system E is still conserved.

Conclusions
In conclusion, this study theoretically investigates the kinetic energy and potential energy of a spring, and kinetic energy mass M with different ratios ms/M. The kinetic energy and potential energy of a spring, and the kinetic energy mass M are numerically simulated. The kinetic energy and potential energy can transform each other, whereas the total energy of the system is conserved. The change behaviors of the spring's kinetic energy are the same as that of the oscillator energy with small ratio ms/M, while kinetic energy of the oscillator gets its maximum before the spring kinetic energy for big ratio ms/M. Moreover, the maximum of the spring kinetic energy is greater than that of the oscillator when the ratio ms/M is big enough.