Summary

General Principles

Doing work on a system changes the system energy:

Δ E_{sys} = W_{ext}

For a system of particles or of extended objects that interact only via friction, $E_{sys} = K + E_{th}$ . In this case,

Δ K + Δ E_{th} = W_{tot}

where $W_{tot}$ is the total work done.

Kinetic energy is an energy of motion: $K = \frac{1}{2} m v^{2}$

Potential energy is stored energy.

Thermal energy is the microscopic energy of moving atoms and stretched bonds. Only extended objects have thermal energy.

Dissipative forces, such as friction and drag, transform macroscopic energy into thermal energy. For friction:

Δ E_{th} = f_{k} Δ s

The work done by a force on a particle as it moves from $s_{i}$ to $s_{f}$ is

W = \int_{s_{i}}^{s_{f}} F_{s} d s = area under the force curve

The work done by a constant force is

W = \vec{F} \cdot Δ \vec{r}

The work done by a spring is

W = - (\frac{1}{2} k {(Δ s_{f})}^{2} - \frac{1}{2} k {(Δ s_{i})}^{2})

where $Δ s$ is the displacement of the end of the spring.

The restoring force of an ideal spring is

(F_{Sp})_{s} = - k Δ s

where k is the spring constant and $Δ s$ is the displacement of the end of the spring from equilibrium.

Power is the rate at which energy is transferred or transformed:

P = d E_{sys} / d t

For a particle with velocity $\vec{v}$ , the power delivered to the particle by force $\vec{F}$ is $P = \vec{F} \cdot \vec{v} = F v cos θ$ .

\vec{A} \cdot \vec{B} = A B cos θ = A_{x} B_{x} + A_{y} B_{y}