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Opus in profectus

Centripetal Force

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Discussion

centripetal

discussion

ac = v2
r

calculus

[diagram of standard position circle]

r    ⇒  x =  +r cos(ωt) y =  +r sin(ωt)
 
v =  dr  ⇒  vx =  rω sin(ωt) vy =  +rω cos(ωt)
dt
a =  dv  ⇒  ax =  rω2 cos(ωt) ay =  rω2 sin(ωt)
dt

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

Note

r2  =  x2  +  y2
r2  =  [+r cos(ωt)]2   +  [+r sin(ωt)]2 
r  =  r

Note

v2  =  vx2  +  vy2
v2  =  [−rω sin(ωt)]2   +  [+rω cos(ωt)]2 
v  =  rω

Note

a2  =  ax2  +  ay2
a2  =  [−rω2 cos(ωt)]2  +  [−rω2 sin(ωt)]2
a  =  rω2

centrifugal

discussion

rotating reference frame