Understanding Work and Energy Changes in AP Physics C: Mechanics

Work (physics definition)

Energy transferred into or out of an object by a force acting through a displacement; if there is no displacement, the force does no work.

Constant-force work (vector form)

For a constant force over a displacement, work is the dot product: $W = \textbf{F} \bullet \triangle \textbf{r}$.

Dot product (in work)

The operation that multiplies the magnitudes of two vectors and the cosine of the angle between them; it makes work depend on alignment between force and displacement.

Constant-force work (scalar form)

$W = F \triangle r \cos\theta$, where $\theta$ is the angle between the force and the displacement.

Parallel component of force (F∥)

The component of a force along the displacement direction; $F_{\bot} = F \cos\theta$, and only this component contributes to work.

Work from the parallel component

For constant force, $W = F_{\bot} \triangle r$.

Positive work

Work done when the force component along the displacement is in the same direction as the displacement; typically increases kinetic energy.

Negative work

Work done when the force component along the displacement is opposite the displacement; typically decreases kinetic energy.

Zero work

Occurs when the displacement is zero or when the force is perpendicular to the displacement ($\cos \theta = 0.$)

Net work (Wnet)

The sum of the works done by all forces acting during the displacement: $W_{net} = \Sigma W_i$.

Work by friction

For kinetic friction opposing motion over distance d, Wf = −fk d (negative because friction opposes displacement).

Normal force does zero work (common case)

For motion along a surface, the normal force is often perpendicular to the displacement, so its work is WN = 0.

Centripetal/normal force in uniform circular motion (work)

A radial force (normal or tension) is perpendicular to the tangential displacement at each instant, so it does zero work even though it changes direction of velocity.

Gravitational force near Earth

Approximated as constant: $\textbf{F}_g = m \textbf{g}$, directed downward.

Work done by gravity (vertical displacement)

For vertical motion with $\triangle y$ positive upward: $W_g = -mg \triangle y$; depends only on vertical change in a uniform gravitational field.

Work by a variable force (differential form)

Over an infinitesimal displacement, $dW = \textbf{F} \bullet d\textbf{r}$.

Work by a variable force (integral form)

Total work along a path: $W = \int \textbf{F} \bullet d\textbf{r}$.

1D work integral (AP Physics C common form)

For motion along $x$ with force component $F_x(x)$: $W = \int_{x_i}^{x_f} F_x(x) \, dx$.

Signed area under an F–x graph

Work equals the signed area between the force-position curve and the x-axis from $x_i$ to $x_f$; above axis gives positive work, below gives negative work.

Hooke’s law (spring force)

For an ideal spring in 1D: $F_x = -kx$, where $k$ is the spring constant and $x$ is displacement from equilibrium.

Work done by a spring (general)

From $x_i$ to $x_f$: $W_{spring} = \int_{x_i}^{x_f} (-k x) \, dx = -\frac{1}{2}k (x_f^2 - x_i^2)$.

Work done by a spring (from equilibrium)

From $x = 0$ to $x$: $W_{spring} = -\frac{1}{2}k x^2$ (negative when compressing/stretching away from equilibrium because spring force opposes displacement).

Work-energy theorem

The net work done on an object equals the change in its kinetic energy: $W_{net} = \triangle K$.

Kinetic energy

Energy of motion: $K = \frac{1}{2}mv^2$.

Stopping distance from constant braking force

Using $W_{net} = \triangle K$ with a constant braking force $F_b$ opposite motion on level ground: $d = \frac{mv_i^2}{2F_b}$.