Step 1: Identify where the unbalanced load actually comes from.
In a two cage drum winding system, when the loaded cage sits at the shaft bottom and the empty cage sits at the top, essentially the entire length of hoisting rope is hanging on the loaded cage's side, while almost none is hanging on the empty cage's side. That difference in the sheer weight of dangling rope, not the coal or men being carried, is what creates a large out of balance torque on the winding drum, especially demanding right at the start of the wind.
Step 2: Think about what would cancel that imbalance as the wind proceeds.
As winding proceeds, the loaded cage's hoisting rope gets shorter (lighter) while the empty cage's hoisting rope gets longer (heavier), so ideally we want something on the loaded cage's side to get correspondingly heavier at the same rate to keep the two sides balanced throughout the wind.
Step 3: Describe how a balancing rope achieves exactly that.
A balancing (tail) rope is hung in a loop beneath both cages, connecting their undersides. As the loaded cage rises, its length of balancing rope increases even as its hoisting rope shortens; as the empty cage descends, its balancing rope shortens even as its hoisting rope lengthens. The combined weight of hoisting rope plus balancing rope on each side therefore stays roughly constant throughout the wind, which is precisely how the static torque on the drum is kept balanced.
\[ \boxed{\text{Balancing rope}} \]