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The present study describes preparation of doped activated carbon (NAC), employing waste orange peel as carbon source, melamine as nitrogen dopant and KOH as activating agent. The prepared NAC samples were textually characterized using the techniques of surface area and pore size analyzer, scanning electron microscopy (SEM), x-ray diffraction (XRD) and Raman spectroscopy. As evident from characterization results, the synthesized NAC materials own porous structure and offers high surface area , and pore volume . Such useful characteristics of NAC indicate its suitability as electrode for supercapacitors. Electrochemical performance of NAC material was evaluated in 6 M KOH aqueous solution, employing the standard electrochemical avenues of analysis. It was found that synthesized NAC sample exhibits high specific capacitance , specific energy and specific power at current load of . The better electrochemical performance of the NAC is probably due to enhanced surface area and availability of nitrogen functional groups. Thus, the porous structure and nitrogen-doping characteristics make NAC a potential electrode material for applications in the field of supercapacitors.

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The motion of a simple pendulum of arbitrary amplitude is usually treated by approximate methods. By using generalized hypergeometric functions, it is however possible to solve the problem exactly. In this paper, we provide the exact equation of motion of a simple pendulum of arbitrary amplitude. A new and exact expression for the time of swinging of a simple pendulum from the vertical position to an arbitrary angular position θ is given by equation (3.10). The time period of such a pendulum is also exactly expressible in terms of hypergeometric functions. The exact expressions thus obtained are used to plot the graphs that compare the exact time period T(θ0) with the time period T(0) (based on simple harmonic approximation). We also compare the relative difference between T(0) and T(θ0) found from the exact equation of motion with the usual perturbation theory estimate. The treatment is intended for graduate students, who have acquired some familiarity with the hypergeometric functions. This approach may also be profitably used by specialists who encounter during their investigations nonlinear differential equations similar in form to the pendulum equation. Such nonlinear differential equations could arise in diverse fields, such as acoustic vibrations, oscillations in small molecules, turbulence and electronic filters, among others.