Centripetal Force

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 acceleration. This shows that acceleration is toward the center at all times (that is, opposite the radius). 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

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