By definition:
#delta W = bb F * d bb r qquad = bb F * (d bb r)/(dt) \ dt = bb F * bb v \ dt#
So it would be helpful to have the velocity vector .
Velocity
#bb F = m bba#
#bb a = a/m(sin omega t bb hat i +cos omega t bb hat j)#
#bb v = int bb a dt = a/m (-1/omega cos omega t bb hat i + 1/omega sin omega t bb hat j) + bb C#
#bb v(0) = bb 0 implies bbC = a/(m omega) bb hat i#
#implies bb v = a/(omega m) (1- cos omega t) bb hat i + sin omega t bb hat j) #
Work
#W = int_0^t a(sin omega tau bb hat i +cos omega tau bb hat j)* a/(omega m) ((1- cos omega tau) bb hat i + sin omega tau bb hat j) \ d tau#
#=a^2/(omega m) int_0^t \ sin omega tau - sin omega tau cos omega tau + sin omega tau cos omega tau \ \ d tau #
#=a^2/(omega m) ( - 1/omega cos omega tau\ )_0^t#
#=a^2/(omega^2 m) ( 1 - cos omega t\ )#
Power
#P = (dW)/(dt) =a^2/(omega^2 m) ( 0 + omega sin omega t\ )#
# =a^2/(omega m) sin omega t \ #