Physics 218: Mechanics Instructor: Dr. Tatiana Erukhimova Lecture 14

Force of gravity: Potential energy function: Spring: Potential energy function:

A particle moves in one dimension under the influence of a single conservative force given by where  is a given constant. Take the potential energy reference to be at x i =0 such that: and calculate the potential energy function U(x).

2D case:

If or then 2 or 3D cases:

Several dimensions: U(x,y,z) Compact notation using vector del, or nabla: Another notation: Partial derivative is taken assuming all other arguments fixed

Geometric meaning of the gradient Direction of the steepest ascent; Magnitude : the slope in that direction Direction of the steepest descent Magnitude : the slope in that direction

Given the potential energy function find the x and y components of the corresponding force.