Question

The pulleys in figure (10-E6) are identical, each having a radius R and moment of inertia I. Find the acceleration of the block M.

Answer 1

According to the question
Mg – T₁ = Ma …(1)
(T₂ – T₁)R = Ia/R  (T₂ – T₁) = Ia/R² …(2)
(T₂ – T₃)R = Ia/R² …(3)
T₃ – mg = ma …(4)
By adding equation (2) and (3) we will get,
(T₁ – T₃) = 2 Ia/R² …(5)
By adding equation (1) and (4) we will get

– mg + Mg + (T₃ – T₁) = Ma + ma …(6)
Substituting the value for T₃ – T₁ we will get
Mg – mg = Ma + ma + 2Ia/R²
a = (M -m)G /(M +m)+2l/R²)