Many natural phenomena we find in our surroundings, are fractals. Studying and learning about fractals in classrooms is always a challenge for both teachers and students. We here show that the sound of musical instruments can be used as a good resource in the laboratory to study fractals. Measurement of fractal dimension which indicates how much fractal content is there, is always uncomfortable, because of the size of the objects like coastlines and mountains. A simple fractal source is always desirable in laboratories. Music serves to be a very simple and effective source for fractal dimension measurement. In this paper, we are suggesting that music which has an inherent fractal nature can be used as an object in classrooms to measure fractal dimensions. To find the fractal dimension we used the box-counting method. We studied the sound produced by different stringed instruments and some common noises. For good musical sound, the fractal dimension obtained is around 1.6882.
In undergraduate classrooms, while teaching chaos and fractals, it is taught as if there is no relation between these two. By using some non linear oscillators we demonstrate that there is a connection between chaos and fractals. By plotting the phase space diagrams of four nonlinear oscillators and using box counting method of finding the fractal dimension we established the chaotic nature of the nonlinear oscillators. The awareness that all chaotic systems are good fractals will add more insights to the concept of chaotic systems.
The equation of state of an ideal collection of bosons in the low-density and high-density regime are found using the method of cluster expansion with Mayer's generating function. The saturation density and the other thermodynamic properties are calculated by the application of Mayer's convergence of the partition function. By calculating the value of saturation density from the singularity of the partition function series, the differences between the Mayer series convergence and the virial series convergence for ideal bosons are also established.
The spin-half Ising model has been used with remarkable success in describing magnetism and phase transition in magnetic materials. Likewise, the lattice gas model, which uses the spin-1 Ising model, was employed to investigate the phase transition in liquid-gas systems. In all the earlier lattice gas models, the kinetic energy of atoms occupying the various lattice sites were not considered for finding the equation of state. In this paper we use the Hamiltonian with kinetic energy and a modified equation of state is obtained. The isotherms of this model show the behavior of real gases at all pressures and they also exhibit phase transition.
Many consider physics to be a highly mathematical oriented subject to study. To break this opinion and also to generate a deep interest in physics, a course on ‘Physics of Music’ can be introduced at any level of a curriculum. We present a simple and practical way of introducing this topic even for school level students. Teachers, along with students, can visualise and feel physics all time throughout the course.
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