Exact solution for the nonlinear pendulum

This paper deals with the nonlinear oscillation of a simple pendulum and presents not only the exact formula for the period but also the exact expression of the angular displacement as a function of the time, the amplitude of oscillations and the angular frequency for small oscillations. This angular displacement is written in terms of the Jacobi elliptic function sn(u;m) using the following initial conditions: the initial angular displacement is different from zero while the initial angular velocity is zero. The angular displacements are plotted using Mathematica, an available symbolic computer program that allows us to plot easily the function obtained. As we will see, even for amplitudes as high as 0.75π (135◦) it is possible to use the expression for the angular displacement, but considering the exact expression for the angular frequency ω in terms of the complete elliptic integral of the first kind. We can conclude that for amplitudes lower than 135◦ the periodic motion exhibited by a simple pendulum is practically harmonic but its oscillations are not isochronous (the period is a function of the initial amplitude). We believe that present study may be a suitable and fruitful exercise for teaching and better understanding the behavior of the nonlinear pendulum in advanced undergraduate courses on classical mechanics.This work was supported by the “Ministerio de Educación y Ciencia”, Spain, under project FIS2005-05881-C02-02, and by the “Generalitat Valenciana”, Spain, under project ACOMP/2007/020

Pleasure, pressure and power: Some contradictions of gendered sexuality

The AIDS epidemic has encouraged public discussion of safer sex, but heterosexual young women have to negotiate sexual relationships with men in situations in which sex is defined largely in terms of men's needs and which lack notions of a positive female sexuality or female desires. Analysis of data from the Women, Risk and AIDS Project is interpreted to show both the range of pressures on young women to engage in sexual practices which are risky, violent or not pleasurable, but also the possibilities for young women to empower themselves in sexual relationships. Women's control over sexual safety is undermined by the dominance of male sexuality and women's compliance in satisfying men's desires. Empowerment is a contradictory and contested process requiring both critical reflection (intellectual empowerment) and the transforming of sexual experiences (experiential empowerment), but some young women are able to put into practice ways of negotiating safe and pleasurable sexual encounters with men