Definition: Work

The work $W$ done by a constant force $F$ over a displacement $d$ is equal to $$W=F_\parallel d=Fd\cos\theta$$Work has units of Joules, where $1\,\unit {J}=1\,\unit{N\cdot m}$. Remember to keep track of sign! If the force and displacement are in the same direction, their product will be positive, and if they point in different directions their product will be negative.

Idea: $F-d$ Graphs

When force $F$ is plotted against distance $d$, the work done between two points is the area under the curve between those two points.

Definition: Work-Energy Theorem

The kinetic energy of a mass $m$ at speed $v$ is equal to $$K=\frac 12 mv^2$$ Note that kinetic energy is a scalar, and is always positive. The work-energy theorem states that given work $W$ acting on an object, that work equals the change in kinetic energy of the object: $$\Delta K=W$$

Definition: Spring Forces

A spring exerts a force $F=-kx$ on an object, where $x$ is the displacement from the relaxed state of the spring. $k$ is called the spring constant, which has units $\unit{N/m}$. The negative sign indicates that when a spring is stretched, it tends to retract, and when a spring is compressed, it tends to expand.

Definition: Potential Energy

Roughly speaking, a conservative force is a force that conserves energy, i.e. it does not result in dissipation of energy to heat or other forms. For a more precise definition, a conservative force is such that the work done by that force is independent of an object's path. Gravity and spring forces are conservative forces, while friction is not. For conservative forces, the potential energy associated with a force is equal to negative the work done by the force over some distance. In other words, $$\Delta P=-W$$ For gravity, potential energy is $P=mgh$ (where $h$ is the height with respect to a reference point), and for springs the potential energy is $P=(1/2)kx^2$ (where $x$ is the stretch/compression of the spring).