03 / CHAPTER 03
Potential energy
Represent conservative interactions with an energy landscape.
ANCHOR EQUATION
Potential force
In symbols: F = −∇U
A conservative force points downhill on its potential-energy landscape.
Potential energy encodes reversible work
A conservative force can trade kinetic energy for a stored quantity whose value depends only on position. Potential energy is useful because it turns repeated force integration into a state difference.
Define the potential by its slope
For a conservative force, define ΔU = −W_force. In one dimension, dU = −F dx, so F = −dU/dx. Near Earth with upward y and constant gravity F_y = −mg; integrate dU = mg dy to get U_g = mgy + constant. The arbitrary constant disappears from differences.
Lifting 2 kg by 3 m increases gravitational potential energy by ΔU = m g Δy = 2(9.81)(3) = 58.9 J. The same amount of work is required by a slow lift if kinetic energy is unchanged.
Not every force has a potential
Friction depends on path and dissipates mechanical energy as thermal energy, so a single-valued U(x) cannot represent it globally. Magnetic force can do no work on a point charge even though it changes momentum direction.
Work it: For a spring with F = −kx, integrate −F dx from x = 0 to x to derive U = ½kx². Evaluate the stored energy for k = 400 N/m and x = 0.05 m.