Understanding Work and Energy Changes in AP Physics C: Mechanics

Vocabulary flashcards covering work done by constant and variable forces, Hooke's law, dot products, integrals, and the work-energy theorem for AP Physics C: Mechanics.

Work (in mechanics)

A measure of how much a force transfers energy into or out of an object through a displacement.

Work Done by a Constant Force (Dot Product Form)

The dot product of force and displacement vectors, defined mathematically as $W = \vec{F} \cdot \Delta\vec{r} = F\,\Delta r\cos(\theta)$, where $F$ is the magnitude of the force, $\Delta r$ is the magnitude of displacement, and $\theta$ is the angle between the force and displacement directions.

Parallel Component of Force ($F_{\parallel}$)

The component of a force aligned with displacement, given by $F_{\parallel} = F\cos(\theta)$; it is the only component of force that contributes to work ($W = F_{\parallel}\,\Delta r$).

Positive Work

Work done when a force component points in the direction of displacement ($\theta < 90^\circ$), transferring energy into kinetic energy and typically speeding the object up.

Negative Work

Work done when a force component points opposite to displacement ($\theta > 90^\circ$), removing energy from kinetic energy and typically slowing the object down.

Zero Work

The condition where no energy transfer occurs, which happens when the force is perpendicular to displacement ($\theta = 90^\circ$) or when the object undergoes zero displacement.

Net Work ($W_{\text{net}}$)

The total work performed on an object by all acting forces, calculated as the scalar sum of the individual works done: $W_{\text{net}} = \sum W_i$.

Work Done by Gravity (Uniform Field)

The work performed by the constant gravitational force near Earth's surface ($F_g = mg$), expressed as $W_g = -mg\Delta y$, which depends only on the vertical displacement $\Delta y$ and is independent of the path taken.

Work Done by a Variable Force (1D)

The total work obtained by integrating the position-dependent force along the direction of motion: $W = \int_{x_i}^{x_f} F_x(x)\,dx$.

Force-Position Graph (Work Interpretation)

A graphical representation where the signed area between the $F_x$ versus $x$ curve and the position axis from $x_i$ to $x_f$ equals the work done; areas above the axis represent positive work and areas below represent negative work.

Hooke's Law

The 1D constitutive equation for an ideal spring, given by $F_x = -kx$, where $k$ is the spring constant and $x$ is the displacement from equilibrium.

Work Done by a Spring Force

The work performed by an ideal spring during deformation between two positions, given by $W_{\text{spring}} = -\frac{1}{2}k(x_f^2 - x_i^2)$, or $W_{\text{spring}} = -\frac{1}{2}kx^2$ when deformed by a magnitude $x$ starting from equilibrium ($x_i = 0$).

Kinetic Energy ($K$)

The translational energy of motion of an object of mass $m$ moving at speed $v$, defined as $K = \frac{1}{2}mv^2$.

Work-Energy Theorem

The theorem establishing that the net work done by all forces acting on an object equals its change in kinetic energy: $W_{\text{net}} = \Delta K = K_f - K_i$.

Stopping Distance Relation

The relation $d = \frac{mv_i^2}{2F_b}$, derived from the work-energy theorem, showing that the stopping distance $d$ of an object with mass $m$ brought to rest by a braking force $F_b$ scales quadratically with initial speed $v_i^2$.