Centripetal force:
a force which acts on a body moving in a circular path and is directed towards the centre around which the body is moving.
Formula:
\[a_{c} =\frac{ v^{2}}{r} \]
Derivation by calculus:
\[r \rightarrow x = +rcos(\omega t ); \: y = +rsin(\omega t) \]
\[v = \frac{dr}{dt} \rightarrow v_{x} = -r\omega \: sin(\omega t) ; v_{y}= -r\omega \, sin(\omega t) \]
\[a = \frac{dv}{dt} \rightarrow a_{x} = -r\omega ^{2} \frac{}{}cos(\omega t) ; a_{y}= -r\omega^{2} \, sin(\omega t) \]
Check out the negative sign in the acceleration. This shows that acceleration is rotated 180° away from the radius; that is, in the opposite direction or toward the center (centripetal). Harder to see is the 90° phase difference between velocity and acceleration.
\[r^{2} = x^{2} + y^{2} \]
\[r^{2} = [+r\omega \: cos(\omega t)]^{2} + [+r\omega \, sin(\omega t)]^{2} \]
\[v^{2} =v _{x}^{2} + v_{y}^{2} \]
\[v^{2}= [- r\omega sin (wt)]^{2} + [+r\omega cos(\omega t)]^{2} \]
\[v = r\omega^{2} \]
\[a^{2}= a_{x}^{2} + a_{y}^{2} \]
\[a^{2}= [-r\omega ^{2} cos(\omega t)]^{2}+ [-r\omega ^{2} sin(\omega t)]^{2} \]
\[a = r\omega ^{2} \]
