Fundamental Principles of Continuum Mechanics

Force Vectors add according to the "Parallelogram Rule" (Law of vector addition) SmartSelect_20230926_133914_Samsung Notes.jpg
1. Commutative Property: $\vec{F_{1}} + \vec{F_{2}} = \vec{F_{2}} + \vec{F_{1}}$
2.

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Assume Force $\vec{F}$ is applied to a rigid body at point $A$ .
Apply equal and opposite forces $\vec{F}$ and $- \vec{F}$ at point $B$ .
$\vec{F}$ at point $A$ cancels $- \vec{F}$ at point $B$ .
Force $\vec{F}$ has been translated from point $A$ to point $B$ .
Thus the point of application of a force can be translated anywhere along its line of action without changing the external reaction forces acting on the body.

Newton's First Law
1. Force Equilibrium $\to \sum_{i} \vec{F_{i}} = \vec{0}$
2. Moment Equilibrium $\to \sum_{i} \vec{M} = \vec{0}$ where $\vec{M} = \vec{r} \times \vec{F}$
Newton's Second Law
1. $\sum \vec{F} = \frac{d}{d t} (m \vec{v}) = m \frac{d V}{d x} \frac{d x}{d t}$
2. If $m$ =constant $\sum \vec{F_{i}} = m \frac{d \vec{v}}{d t}$
3. Units of force: $[M] [\frac{d v}{d t}] = [k g] [\frac{m}{s^{2}}] \to$ 1 Newton (1N) = $\frac{k g \cdot m}{s^{2}}$
Newton's Third Law
1. $\vec{F_{i j}} = - \vec{F_{i j}}$
2. Reaction forces
Law of Gravitation
1. $\frac{R^{3}}{T^{2}} = c$
2. $F = m (\frac{v^{2}}{r})$
  $\frac{2 π r}{T} = v$
  $F = \frac{m}{r} {(\frac{2 π r}{T})}^{2}$
  $F = 4 π^{2} \frac{m \cdot r}{T^{2}}$
  $\frac{G M m}{r^{n}} = m \frac{4 π^{2} r}{T^{2}}$
  $G M = 4 π^{2} \frac{r^{n + 1}}{T^{2}}$
  $\frac{r^{3}}{T^{2}} = \frac{G M}{4 π^{2}} \to$ Kepler's Value of $c = \frac{G M}{4 π^{2}}$
3. $F_{g} = \frac{G M_{E} m}{r_{E}^{2}}$
  $\frac{F_{g}}{m} = \frac{G M_{E}}{r_{E}^{2}} \leftarrow g = \frac{G M_{E}}{r_{E}^{2}}$
  Acceleration due to gravity: $W = m g \leftarrow g = 9.81 \frac{m}{s^{2}}$