A particle is moving in a circle of radius r under the action of a force F=αr² which is directed towards the center of the circle. Total mechanical energy (kinetic energy + potential energy) of the particle is (take potential energy = 0 for r = 0)

The force F is providing centripetal acceleration to the particle
The kinetic energy is given as 1/2mv²
F = mv²/r = αr²
multiplying both sides by r and dividing by 2 ⇒ 1/2mv² = αr³/2
The potential energy is work done by force F from moving the mass from r=0 to r=r
U = ∫₀ʳ Fdr
U = ∫₀ʳ αr²dr
U = αr³/3
Total energy = K.E. + P.E. = αr³/2 + αr³/3 = 5αr³/6
Hence correct answer is option A.