A particle lis constrained to move in a circular path of radius r = 6m. Its velocity varies with time according to the relation v = 2t (m/s). Determine its
(i) centripetal acceleration,
(ii) tangential acceleration,
(iii) instantaneous acceleration at (a) t = 0 sec. and
(b) t = 3 sec.

Explanation

(a) At = 0, v = 0, thus `a_(r) = 0`
but `(dv)/(dt) = 2 " thuse " a_(t) = 2 " "m//s^(2)`
and a = `sqrt(a_(t)^(2) + a_(r )^(2)) = 2 " "m//s^(2)`
(b) At t = 3 sec. v = 6 m/s
so `a_(r ) = (v^(2))/(r ) = ((6)^(2))/(6) = m//s^(2)`
and `a_(t) = (dv)/(dt) = 2 m//s^(2)` Therefore,
a = a = `sqrt(2^(2) + 6^(2)) = sqrt(40) " "m//s^(2)`