An approximation scheme to obtain the period for large amplitude oscillations of a simple pendulum is analysed and discussed. The analytical approximate formula for the period is the same as that suggested by Hite (2005 Phys. Teach. 43 290), but it is now obtained analytically by means of a term-by-term comparison of the power-series expansion for the approximate period with the corresponding series for the exact period. Approximation for a large-angle simple pendulum period

The harmonic balance method is used to construct approximate frequency-amplitude relations and periodic solutions to an oscillating charge in the electric field of a ring. By combining linearization of the governing equation with the harmonic balance method, we construct analytical approximations to the oscillation frequencies and periodic solutions for the oscillator. To solve the nonlinear differential equation, firstly we make a change of variable and secondly the differential equation is rewritten in a form that does not contain the squareroot expression. The approximate frequencies obtained are valid for the complete range of oscillation amplitudes and excellent agreement of the approximate frequencies and periodic solutions with the exact ones are demonstrated and discussed.

The Doppler effect is a phenomenon which relates the frequency of the harmonic waves generated by a moving source with the frequency measured by an observer moving with a different velocity from that of the source. The classical Doppler effect has usually been taught by using a diagram of moving spheres (surfaces with constant phase) centred at the source. This method permits an easy and graphical interpretation of the physics involved for the case in which the source moves with a constant velocity and the observer is at rest, or the reciprocal problem (the source is at rest and the observer moves). Nevertheless it is more difficult to demonstrate, by this method, the relation of the frequencies for a moving source and observer. We present an easy treatment where the Doppler formulae are obtained in a simple way. Different particular cases will be discussed by using this treatment.

In their comment, Qing-Yin and Pei derived another analytical approximate expression for the large-angle pendulum period, which they compare with other expressions previously published. Most of these approximate formulas are based on the approximation of the original nonlinear differential equation for the simple pendulum motion. However, we point out that it is possible another procedure to obtain an approximate expression for the period. This procedure is based on the approximation of the exact period formula-which is expressed in terms of a complete elliptic integral of the first kind-instead of the approximation of the original differential equation. This last procedure is used, for example, by Carvalhaes and Suppes using the arithmeticgeometric mean.

The analytical approximate technique developed by Wu et al for conservative oscillators with odd nonlinearity is used to construct approximate frequency-amplitude relations and periodic solutions to the dynamically shifted oscillator. This nonlinear oscillator is described by an equation of motion which includes a linear restoring force and an anti-symmetric, constant force which is a nonlinear force depending only upon the sign of the displacement. By combining Newton's method with the method of harmonic balance, analytical approximations to the oscillation frequency and periodic solutions are constructed for this oscillator and the approximate periods obtained are valid for the complete range of oscillation amplitudes. Excellent agreement of the approximate frequencies and periodic solutions with the exact ones are demonstrated and discussed and the results reveal that this technique is very effective and convenient for solving this class of conservative nonlinear oscillatory systems.