Why does centrifuge tubes must be equay heavy and placed at equal intervals in the rotor?

Answer 1

Because weight is directly related to mass and one must maintain a balanced arrangement or else the centrifuge would bounce around on the table. (The "false" force of inertia would cause this bouncing.)

Answer 2

Centrifugal force is proportional to its revolutions squared times its mass.

If a mass is not balanced when revolving, the force that will rip in to the apparatus is squared compared to static forces brought about by gravity or such.

Answer 3

For balance. You would not want such a expensive machine to wobble out of a smooth rotation, would you?