Tue May 08, 2012 By:

why centripetal acceleration is v^2/r how we can justify it

Expert Reply
Tue May 08, 2012
  1. Consider v at point (x,y) in a circle about the origin, as shown.


The velocity vector v has an angle to the -x axis that is the complement of q , hence

vx = -vsinq and vy = vcosq . (equations 1)

But sin and cos of q are y/R and x/R, so

vx = -vy/R and vy = vx/R. (equations 2)

Find the rates of change of these and we have ax and ay.
Note that v and R are constant, and the rates of change of y and x are vy and vx, so

ax = -vvy/R and ay = vvx/R. (equations 3)

Now substitute equations 2 into equations 3 and get 
ax = -v2x/R2 and ay = -v2y/R2,
and from a = (ax2 + ay2)1/2 and x2 + y2 = R2 we find

a = v2/R.  Velocity is related to angular velocity w by v = wR if w is in rad/s, so a = w2R is an alternative equation.

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