Long pendulums in gravitational gradients

Previous results for long pendulums above a spherical Earth are generalized for arbitrary non-uniform gravitational fields in the limit of small oscillation. As is the case for the previous results, gravitational gradients are multiplied by the length of the string even though the string is assumed massless. The effect is shown to arise from the constraint on the motion imposed by the string. The significance of these results for real gradients is discussed. (letters and comments)