Simple Nature - Light and Matter

increasing most rapidly, and it tells you how rapidly U is increasingin that direction. For instance, sperm cells in plants and animalsfind the egg cells by traveling in the direction of the gradient of theconcentration of certain hormones. When they reach the locationof the strongest hormone concentration, they find their destiny. Interms of the gradient, the force corresponding to a given interactionenergy is F = −∇U.Force exerted by a spring example 78In one dimension, Hooke’s law is U = (1/2)kx 2 . Suppose wetether one end of a spring to a post, but it’s free to stretch andswing around in a plane. Let’s say its equilibrium length is zero,and let’s choose the origin of our coordinate system to be at thepost. Rotational invariance requires that its energy only dependon the magnitude of the r vector, not its direction, so in two dimensionswe have U = (1/2)k|r| 2 = (1/2)k ( x 2 + y 2) . The forceexerted by the spring is thenF = −∇U= − ∂U ∂U ˆx −∂x ∂y ŷ= −kx ˆx − kyŷ .The magnitude of this force vector is k|r|, and its direction is towardthe origin.This chapter is summarized on page 948. Notation and terminologyare tabulated on pages 941-942.Section 3.4 Motion In Three Dimensions 217