• For two coordinate systems related by $\vec r'=\vec r-\vec R$ then $\vec F_\text{inertial}=-m\ddot{\vec R}$
• For rotating coordinate system $\dot {\vec e'_i}=\vec\omega\times\vec e'_i$
• $\dot {\vec A}|S=\dot{\vec A}|{S'}+\vec\omega\times\vec A$
• General expression for the inertial force in an accelerating and rotating frame:

$$ \vec F_\text{inertial}=-m\ddot{\vec R}|_S-m(\dot{\vec\omega}\times\vec r')-2m({\vec\omega}\times\vec v')-m(\vec\omega\times[\,\vec\omega\times\vec r'\,]) $$

• $\vec F_\text{cor}=-2m(\vec\omega\times\vec v')$
• $\vec F_\text{cen}=-m(\vec\omega\times[\,\vec\omega\times\vec r'\,])$
• $\vec F_\text{cor}\perp\vec \omega$
• In the context of the rotation of the Earth the centrifugal force is responsible for a $0.2\%$ modification to $g$ and a static pendulum will not be vertical - $\alpha_max\approx0.1\degree$
• To first approximation, a particle falling from a height $h$ above the surface of the earth at latitude $\lambda$ will deviate by a distance $\Delta x=\frac 13 g\omega \left( \frac{2h}g \right)^\frac 32 \cos\lambda$