Introduction:
In this problem, we will analyze the motion of a particle of mass m moving in a horizontal circle of radius r under the influence of a centripetal force equal to -k/r^2, where k is a constant. We will determine the total energy of the particle and provide a detailed explanation.

Analysis:
To analyze the motion of the particle, we need to consider the forces acting on it. Since the particle is moving in a circle, there must be a centripetal force acting towards the center. In this case, the centripetal force is given by -k/r^2, where k is a constant.

Centripetal Force:
The centripetal force is given by the equation Fc = m*v^2/r, where Fc is the centripetal force, m is the mass of the particle, v is its velocity, and r is the radius of the circle. In this case, we have Fc = -k/r^2.

Equating Centripetal Forces:
Equating the given centripetal force with the general equation, we have -k/r^2 = m*v^2/r. Simplifying this equation, we get v^2 = -k(r^3/m).

Kinetic Energy:
The kinetic energy of the particle is given by the equation KE = (1/2)*m*v^2. Substituting the value of v^2 from the previous equation, we have KE = (1/2)*m*(-k(r^3/m)). Simplifying this equation, we get KE = -k(r^3/2m).

Potential Energy:
The potential energy of the particle depends on the nature of the force acting on it. In this case, the force is conservative, so we can define a potential energy function. However, since the force is not explicitly given, we cannot determine the potential energy function.

Total Energy:
The total energy of the particle is the sum of its kinetic energy and potential energy. Since we do not have the potential energy function, we cannot determine the exact value of the total energy. However, we can define it as the sum of the kinetic energy and an unknown potential energy function.

Conclusion:
In conclusion, the total energy of the particle moving in a horizontal circle under a centripetal force of -k/r^2 is the sum of its kinetic energy and an unknown potential energy function. The kinetic energy is given by -k(r^3/2m), but the potential energy cannot be determined without knowing the exact nature of the force.