AP Physics C Mechanics — Potential Energy & Energy Conservation (Unit 3)

Kinetic Energy (K)

Energy an object has due to motion; a scalar depending on mass and speed: $K = \frac{1}{2}mv^2$.

Work–Kinetic Energy Theorem

Net work done by all forces equals the change in kinetic energy: $W_{net} = \triangle K = K_f - K_i$.

Net Work (W_net)

Total work done by all forces acting on an object; only net work determines ΔK.

Work (constant force, parallel)

For a constant force parallel to displacement: W = Fd.

Work (general definition)

For forces varying with position/direction: $W = \int \vec{F} \, \bullet \, d\text{r}$; only the component of force along displacement contributes.

Area Under an F–x Graph

For 1D motion, work equals the area under the force-versus-position curve between two x-values.

Speed in Kinetic Energy

$K$ uses speed squared ($v^2$), so $K$ is nonnegative even if velocity is negative.

Potential Energy (U)

Energy associated with a system’s configuration (interactions between objects), especially useful for conservative forces.

Conservative Force

A force whose work between two points depends only on initial and final positions (path independent), allowing a potential energy function U to be defined.

Non-Conservative Force

A force whose work depends on the path (e.g., kinetic friction, air drag), typically changing mechanical energy into thermal/internal energy.

Mechanical Energy (E_{mech})

Sum of kinetic and potential energy: E_mech = K + U.

Conservation of Mechanical Energy

If only conservative forces do work, mechanical energy stays constant: $K_i + U_i = K_f + U_f$.

Energy with Non-Conservative Work

When non-conservative forces do work: $\triangle K + \triangle U = W_{nc}$ (equivalently, $\triangle E_{mech} = W_{nc}$).

Gravitational Potential Energy (near Earth)

In a uniform gravitational field: U_g = mgh (h measured from a chosen reference level).

Change in Near-Earth Gravitational Potential

$\triangle U_g = mg(h_f - h_i) = mg\triangle h$ (only differences matter physically).

Gravitational Potential Energy (universal)

For variable $g$ at large distances: $U_g(r) = -\frac{GMm}{r}$, with $U \rightarrow 0$ as $r \rightarrow \infty$.

Zero Level of Potential Energy

The reference point where U is defined as zero; can be chosen for convenience as long as you use it consistently.

Hooke’s Law (spring force)

Ideal spring force is restoring: F_s = −kx, where x is displacement from equilibrium.

Elastic (Spring) Potential Energy

Energy stored in a deformed ideal spring: $U_s = \frac{1}{2}kx^2$ (nonnegative; typically zero at $x = 0$).

Path Independence (conservative criterion)

For a conservative force, work from A to B is the same for any path taken between A and B.

Closed-Loop Work Test

A force is conservative if the work over any closed path is zero: $\oint \vec{F} \, \bullet \, d\text{r} = 0$.

Conservative Work–Potential Relation

Work done by a conservative force equals negative change in potential energy: $W_{cons} = -\triangle U$.

Force from Potential Energy (1D)

In one dimension, conservative force relates to U(x) by F_x = −dU/dx (slope of U gives force direction/magnitude).

System Choice (internal vs external)

Choosing the system determines whether interactions appear as potential energy (internal) or as external work; don’t double-count the same interaction as both U and W.

Normal Force/Tension and Work

Normal force or tension does work only if it has a component along displacement; often zero when perpendicular to motion (e.g., normal on a surface, tension in an ideal pendulum).