Conservative forces and potential energy

Conservative forces

A force is conservative if the work it does between two points depends only on the initial and final positions, not on the path taken. Gravity near Earth, ideal spring forces, and universal gravitation are conservative forces. Kinetic friction is not conservative because the work it does depends on path length.

For a conservative force,

$W_{conservative}=-Delta U$

where $U$ is potential energy. This means when a conservative force does positive work, potential energy decreases. When it does negative work, potential energy increases.

Path independence

Imagine lifting a book from the floor to a shelf. If gravity is the only force considered, the work done by gravity depends only on the change in height, not on whether the book moved straight up or along a curved path. That is path independence.

For gravity near Earth's surface,

$U_g=mgy$

so

$Delta U_g=mgDelta y$

The work done by gravity is

$W_g=-mgDelta y$

If the book moves upward, gravity does negative work and potential energy increases. If the book falls downward, gravity does positive work and potential energy decreases.

Spring potential energy

An ideal spring obeys Hooke's law:

$F_s=-kx$

The associated potential energy is

U_s=rac{1}{2}kx^2

This energy is smallest at equilibrium, where $x=0$. Stretching or compressing the spring increases potential energy.

Force from potential energy

In one dimension, a conservative force is related to potential energy by

F(x)=-rac{dU}{dx}

The negative sign means force points in the direction of decreasing potential energy.

For a spring,

U(x)=rac{1}{2}kx^2

so

ight)=-kx$$ For gravitational potential energy near Earth, $$U(y)=mgy$$ so $$F_y=-rac{dU}{dy}=-mg$$ ## Potential energy reference level Potential energy values depend on a chosen zero level. Only changes in potential energy are physically meaningful in most mechanics problems. You can choose $U=0$ at the floor, at a table, at equilibrium, or at infinity, depending on convenience. Changing the zero of potential energy does not change forces because force depends on the derivative of $U$, and the derivative of a constant is zero. ## Energy diagrams Potential energy graphs help predict motion. If total mechanical energy is $E$, then $$K=E-U(x)$$ Since kinetic energy cannot be negative, motion is allowed only where $$Egeq U(x)$$ Points where $E=U(x)$ are turning points because $K=0$ and speed is zero. Equilibrium occurs where $$rac{dU}{dx}=0$$ A minimum of $U$ is stable equilibrium. A maximum is unstable equilibrium. ## Non-conservative forces Non-conservative forces, such as kinetic friction or air resistance, transform mechanical energy into thermal energy, sound, or other forms. They cannot generally be represented by a single potential energy function. ## The big idea Conservative forces are special because they allow potential energy to be defined. Work by a conservative force equals the negative change in potential energy. Forces can be recovered from potential energy by taking a negative derivative. Potential energy diagrams provide a powerful way to understand motion, equilibrium, and turning points.