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?

Solution

Physics sir · 4 steps

Take one string at a time: keep each length constant, then chain the pulley accelerations downwards.

1. Top string. Take every acceleration as positive downwards, measured from the ground. Call the middle and lower pulleys P₂ and P₃, with accelerations b₂ and b₃. The string over the fixed pulley joins m₁ to P₂, and its length is fixed, so a₁ + b₂ = 0, which gives b₂ = -a₁.
2. Middle string. This string runs over P₂, from m₂ to P₃. Each side's length is measured from P₂, so (a₂ - b₂) + (b₃ - b₂) = 0. That gives b₃ = 2b₂ - a₂ = -2a₁ - a₂.
3. Bottom string. This string runs over P₃, from m₃ to m₄. Both sides are measured from P₃, so (a₃ - b₃) + (a₄ - b₃) = 0, which gives a₃ + a₄ = 2b₃.

   a₃ + a₄ = 2(-2a₁ - a₂) ⇒ 4a₁ + 2a₂ + a₃ + a₄ = 0

4. So. The constraint relation is 4a₁ + 2a₂ + a₃ + a₄ = 0, which is option (a).

The idea to keep. For pulley chains, write one length constraint per string, measuring each side from its own pulley. A movable pulley's acceleration then feeds into the next string, doubling at every level.

Answer from the NTA final answer key. Solution by PrepWiser.