Centripetal Force Discussion

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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 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

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