Final answer:
The rank of tensions T_a, T_b, T_c, T_d depends on the horizontal and vertical components that must balance the circular motion forces. Tensions would be larger for more horizontal wires and equal if angles are the same.
Step-by-step explanation:
To determine the rank of tensions (Ta, Tb, Tc, Td) experienced by particles moving in horizontal circles on a tabletop, it is critical to consider both the horizontal and vertical components of these tensions. Since the particles are moving at the same speed but possibly with different angles of their tension vectors, different tensions in the wires will result if the angles are not identical.
The horizontal components of the tensions must cancel out for the particles to move in horizontal circles. This means that if one tension is pulling more horizontally than another, it must be larger to provide the necessary centripetal force. Likewise, the sum of the vertical components must balance the weight of the object being held in circular motion.
If tensions Ta, Tb, Tc, and Td were associated with wires having different angles to the horizontal, the tension in the most horizontal wire would be the greatest, and the tension in the most vertical wire would be the smallest. Without the specific angles or additional information provided in the question, a direct comparison cannot be made. However, it is noted that both tensions would be larger if both wires were more horizontal, and they would be equal if and only if the angles on either side are the same.