Three weightless pulleys with radii R, 2R and 3R are concentrically fastened together to make a single triple pulley system and mounted on an axis on which the triple pulley can rotate without friction (figure). The threads are light and they do not slip on the pulley. We suspended masses 8m, m and 2m as shown. What is the acceleration $(i n m / s^{2})$ of mass 8m ? $(g = 10 m / s^{2})$

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answer is 4.

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Detailed Solution

A system of pullies is massless, ${(τ_{n e t})}_{0} = 0$

$T_{1} R + T_{3} (3 R) - T_{2} (2 R) = 0 T_{1} + 3 T_{3} = 2 T_{2}____________(1) 8 m g - T_{1} = 8 m (R α) T_{1} = 8 m (g - R α)_________(2) S i m i l a r l y, T_{3} = 2 m (g - 3 R α)_________(3) T_{2} = m (g + 2 R α)_________(4)$

Substituting (2), (3), (4) in (1)
$8 m (g - R α) + 6 m (g - 3 R α) = 2 m (g + 2 R α)$
$12 g = 30 R α \Rightarrow R α = \frac{12 g}{30} = \frac{120}{30} = 4 m s^{- 2}$