Introduction:
In this problem, we are given that a particle of mass m is moving in a horizontal circle of radius r under the influence of a centripetal force. The magnitude of the centripetal force is given by -k/r², where k is a constant.

Centripetal Force:
The centripetal force is the force that keeps an object moving in a circular path. It always acts towards the center of the circle and is responsible for changing the direction of the object's velocity. In this case, the centripetal force is given by -k/r², which means it is directed towards the center of the circle.

Calculating the Total Mechanical Energy:
The total mechanical energy of a particle is the sum of its kinetic energy and potential energy. In this problem, since the particle is moving in a horizontal circle, there is no change in height and therefore no potential energy. Hence, the total mechanical energy is equal to the kinetic energy of the particle.

The kinetic energy of a particle is given by the formula KE = 1/2 * m * v², where m is the mass of the particle and v is its velocity. In this case, the velocity of the particle is determined by the centripetal force.

Since the centripetal force is given by -k/r², we can equate it to the formula for centripetal force, which is mv²/r. By equating the two equations, we get:

-k/r² = mv²/r

Simplifying this equation, we find:

v² = -k/m

Taking the square root of both sides, we get:

v = √(-k/m)

Substituting this value of v into the formula for kinetic energy, we find:

KE = 1/2 * m * (√(-k/m))²

Simplifying this equation, we get:

KE = 1/2 * m * (-k/m)

Cancelling out the m terms, we find:

KE = -1/2 * k

Hence, the total mechanical energy is given by -1/2 * k.

Conclusion:
The total mechanical energy of the particle moving in a horizontal circle under the influence of a centripetal force equal to -k/r² is equal to -1/2 * k. This means that the particle has a constant amount of mechanical energy throughout its motion, which is negative and half of the constant k.