General Principles of Engineering Mechanics

Engineering Mechanics is an applied science describing and predicting conditions of rest or motion of bodies influenced by forces.
The field is categorized into:
- Rigid bodies: Further divided into Statics and Dynamics.
- Deformable bodies.
- Fluids.

Length: Defines position and geometric properties; associated with a point $P$ and its coordinates relative to an origin.
Time: Required to specify when an event occurs.
Mass: Characterizes a body's resistance to translational motion changes and its response to gravitational attraction.
Force: Vector quantity representing the action of one body on another, defined by magnitude, direction, and point of application. It can result from contact, gravity, or magnetic attraction.
Particle: An object with mass concentrated at a point, having zero volume and negligible dimensions.
Rigid Body: A body where deformation is negligible; the distance between any two points remains constant. In Statics, bodies are assumed rigid unless specified otherwise.
Newtonian Mechanics: Assumes length, time, and mass are absolute and independent concepts.

Scalars: Quantities associated with magnitude only (e.g., $mass$ , $density$ , $volume$ , $time$ , $energy$ ).
Vectors: Quantities defined by magnitude and direction (e.g., $force$ , $displacement$ , $velocity$ , $acceleration$ ).
Velocity ( $\mathbf{v}$ ): The time rate of change of position ( $\mathbf{v} = \frac{d\mathbf{r}}{dt}$ ).
Acceleration ( $\mathbf{a}$ ): The time rate of change of velocity ( $\mathbf{a} = \frac{d\mathbf{v}}{dt}$ ).
Static Equilibrium: A state where $\mathbf{a} = 0$ . If velocity is also zero, the position remains constant.

First Law: An object remains at rest or moves with constant speed in a straight line unless acted upon by an unbalanced force ( $\sum \mathbf{F} = 0$ ).
Second Law: The acceleration of a particle is proportional to the vector sum of forces acting on it ( $\mathbf{F} = m\mathbf{a}$ ). Static equilibrium occurs when the resultant force is zero.
Third Law: Mutual forces of action and reaction between two particles are equal in magnitude, opposite in direction, and collinear.

The force between two particles is calculated as: $F = \frac{GMm}{r^2}$
Universal constant of gravitation: $G = 6.673 \times 10^{-11}\,m^3/kg \cdot s^2$
Weight ( $W$ ): The gravitational force on a mass $m$ at sea level, expressed as $W = mg$ .
Gravity constants: $g = 9.81\,m/s^2$ or $32.2\,ft/s^2$ .
Specific weight ( $\gamma$ ) to density ( $\rho$ ) relationship: $\gamma = \rho g$ .

Common Prefixes:
- Giga ( $G$ ): $10^9$
- Mega ( $M$ ): $10^6$
- Kilo ( $k$ ): $10^3$
- Centi ( $c$ ): $10^{-2}$
- Milli ( $m$ ): $10^{-3}$
- Micro ( $\mu$ ): $10^{-6}$
- Nano ( $n$ ): $10^{-9}$
Rules for Units:
- Prefixes should generally appear only in the numerator (except for $kg$ ).
- Multiplication of units is denoted with a dot or dash (e.g., $N \cdot m$ or $N-m$ ).
- Exponentiation applies to both the unit and the prefix ( $mm^2 = (mm)^2$ ).