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Centripetal acceleration and force formula

Consider an object of mass m moving with constant angular velocity (w) and constant speed (v) in a circle of radius r with centre O.
The mass m moves from P to Q in a time t.
Change in velocity parallel to PO = vsinq - 0
Change in velocity perpendicular to PO = vcosq - v
When Q is very close to P sinq is close to q in radians and cosq tends to 1 and so:

Change in velocity along PO = vq - 0 = vq
Change in velocity perpendicular to PO = v - v = 0
Therefore acceleration along PO = vq/t = vw= v2/r = w2r

Centripetal acceleration (a) = v2/r = q2r
Centripetal force (F) = mv2/r = mw2r
© Keith Gibbs 2011