## Statics

• Statics is the branch of mechanics that is concerned with the analysis of loads (force and torque, or "moment") acting on physical systems that do not experience an acceleration (a=0), but rather, are in static equilibrium with their environment. When in static equilibrium, the acceleration of the system is zero and the system is either at rest, or its center of mass moves at constant velocity. The application of Newton's second law to a system gives:

Where bold font indicates a vector that has magnitude and direction. F is the total of the forces acting on the system, m is the mass of the system and a is the acceleration of the system. The summation of forces will give the direction and the magnitude of the acceleration will be inversely proportional to the mass. The assumption of static equilibrium of a = 0 leads to:

The summation of forces, one of which might be unknown, allows that unknown to be found. Likewise the application of the assumption of zero acceleration to the summation of moments acting on the system leads to:

Here, M is the summation of all moments acting on the system, I is the moment of inertia of the mass and α = 0 the angular acceleration of the system, which when assumed to be zero leads to:

The summation of moments, one of which might be unknown, allows that unknown to be found. These two equations together, can be applied to solve for as many as two loads (forces and moments) acting on the system.

From Newton's first law, this implies that the net force and net torque on every part of the system is zero. The net forces equaling zero is known as the first condition for equilibrium, and the net torque equaling zero is known as the second condition for equilibrium. See statically indeterminate.

A scalar is a quantity which only has a magnitude, such as mass or temperature. A vector has a magnitude and a direction. There are several notations to identify a vector, including:

Vectors are added using the parallelogram law or the triangle law. Vectors contain components in orthogonal bases. Unit vectors i, j, and k are, by convention, along the x, y, and z axes, respectively.

${\displaystyle {\textbf {F}}=m{\textbf {a}}\,.}$
${\displaystyle {\textbf {F}}=0\,.}$
${\displaystyle {\textbf {M}}=I\alpha =0\,.}$
${\displaystyle {\textbf {M}}=0\,.}$
F = the force applied
d = the perpendicular distance from the axis to the line of action of the force. This perpendicular distance is called the moment arm.
${\displaystyle {\textbf {M}}_{O}={\textbf {r}}\times {\textbf {F}}}$
• A bold faced character V
• An underlined character V
• A character with an arrow over it ${\displaystyle {\overrightarrow {V}}}$.
• Beer, F.P. & Johnston Jr, E.R. (1992). Statics and Mechanics of Materials. McGraw-Hill, Inc.
• Beer, Johnston, and Eisenberg (2009). Vector Mechanics for Engineers: Statics, 9th Ed. , McGraw Hill.
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