Thermal noise limitations in torsion pendulum experiments

Statistical estimates of thermal fluctuations of a torsion pendulum used to detect weak forces are re-examined by a more general approach with special reference to gravitational experiments. Thermal limitations on the measurement of the displacement of a pendulum and on the measurement of its period are both studied. In the study of the uncertainty of displacement due to thermal noise it is found that the effect has been underestimated in some experiments on gravitation if the displacement is measured in a discrete way; it is also found that a continuous measurement of the displacement in a non-equilibrium state of a pendulum can depress the thermal noise level greatly; in some special cases it can reduce it by nearly a factor of Q and the formula of Braginsky (1977) and others is re-interpreted. In the study of the uncertainty of the period due to thermal noise, it is found that there is an optimum measurement time for which the uncertainty due to thermal noise in the measurement of the period is a minimum.