In the arrangement shown in figure a₁, a₂, a₃ and a₄ are the accelerations of masses m₁, m₂, m₃ and m₄ respectively. Which of the following relation is true for this arrangement?

Idea: work down the cascade. Each string over a pulley ties its two ends to twice the acceleration of that pulley, and each pulley hangs from the string above it.

Take downward as positive throughout.

Top string, over the fixed pulley: one end holds m₁, the other holds pulley 2. A fixed pulley just reverses the sign, so

a_P2=-a₁.

Second string, over the movable pulley 2: its ends hold m₂ and pulley 3, and for a movable pulley the two ends average to the pulley's own acceleration:

a₂+a_P3=2a_P2=-2a₁, so a_P3=-2a₁-a₂.

Third string, over the movable pulley 3: its ends hold m₃ and m₄:

a₃+a₄=2a_P3=-4a₁-2a₂.

Collecting everything on one side,

4a₁+2a₂+a₃+a₄=0.

The doubling at each stage is what produces the 4,2,1,1 pattern, and the equal coefficients on a₃ and a₄ are forced by their sharing one string.