Statics Equilibrium and Free-Body Diagrams Cheat Sheet

Key Facts

• Translational equilibrium requires $\sum F_x = 0$, $\sum F_y = 0$, and $\sum F_z = 0$ when using three-dimensional coordinates.
• Rotational equilibrium requires $\sum \tau = 0$ about any chosen pivot or axis for a rigid body in static equilibrium.
• Torque magnitude is $\tau = rF\sin\theta$, where $r$ is the distance from the pivot to the force application point and $\theta$ is the angle between $\vec{r}$ and $\vec{F}$.
• A force produces no torque about a pivot if its line of action passes through the pivot, so $\tau = 0$ for that force.
• The weight of an object is $W = mg$ and acts vertically downward through the center of mass.
• Static friction satisfies $0 \le f_s \le \mu_s N$, while maximum static friction is $f_{s,\max} = \mu_s N$.
• Kinetic friction has magnitude $f_k = \mu_k N$ and acts opposite relative sliding motion.
• For an incline at angle $\theta$, weight components are $mg\sin\theta$ parallel to the plane and $mg\cos\theta$ perpendicular to the plane.

Vocabulary

Static equilibrium

A condition in which an object has zero linear acceleration and zero angular acceleration, so $\sum \vec{F} = \vec{0}$ and $\sum \vec{\tau} = \vec{0}$.

Free-body diagram

A diagram that isolates one object and shows every external force acting on it with correct directions and points of application.

Torque

The rotational effect of a force about a point or axis, calculated by $\vec{\tau} = \vec{r} \times \vec{F}$.

Line of action

The straight line extending through a force vector, used to determine the perpendicular lever arm for torque.

Normal force

A contact force perpendicular to a surface that prevents objects from passing through one another.

Coefficient of static friction

The dimensionless constant $\mu_s$ that sets the maximum static friction force by $f_{s,\max} = \mu_s N$.