Force from a Potential Energy Curve

Potential energy stores the work done against a force, so it makes sense that force and energy are tied together through a derivative. If U(x) rises steeply as x increases, the force pushes back the other way, toward lower U, and it pushes hard. If U(x) is flat, there is no force at all. That's the whole idea behind F = -dU/dx: force always points downhill on the energy landscape, and its size matches the steepness of that hill.

Mechanically, you just differentiate U(x) with respect to position and flip the sign. A positive slope in U gives a negative F, meaning the force points in the negative x direction, back toward lower energy. A negative slope gives a positive F. Where the slope is zero, dU/dx = 0, so F = 0, and the system sits at equilibrium. You can tell whether that equilibrium is stable or unstable by looking at the curvature: a local minimum in U (a valley) means any small push creates a restoring force back to the bottom, so it's stable. A local maximum (a hill) means any small push creates a force that shoves the system further away, so it's unstable.

This connects directly to turning points, where kinetic energy hits zero and all the system's energy is potential. On an energy diagram, the total mechanical energy is a horizontal line, and the system can only exist where that line sits at or above U(x). Where the horizontal line crosses the U(x) curve marks a turning point, the edge of the system's allowed motion, where it momentarily stops and reverses. This is the same graph used in Unit 3's energy conservation problems, now read for force and motion limits instead of just energy values.