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