If the pulleys shown in the diagram are smoot | vedclass

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(B) Let the tension in the string connected to the $4 \, kg$ block be $T$. Since the pulley is massless,the tension in the string connected to the $8

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$a_1 = 2 a_2$

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(B) Let the tension in the string connected to the $4 \, kg$ block be $T$. Since the pulley is massless,the tension in the string connected to the $8 \, kg$ block will be $2T$ due to the movable pulley arrangement.For the $8 \, kg$ block:$2T = 8 a_2 \implies T = 4 a_2 \dots (i)$For the $4 \, kg$ block:$4g - T = 4 a_1 \dots (ii)$Using the constraint relation for the movable pulley:If the $8 \, kg$ block moves by a distance $x$,the movable pulley moves by $x$,and the string length constraint implies that the $4 \, kg$ block moves by $2x$. Therefore,the accelerations are related by $a_1 = 2 a_2$.Substituting $T = 4 a_2$ from $(i)$ into $(ii)$:$4g - 4 a_2 = 4 a_1$$g - a_2 = a_1$However,based on the kinematic constraint of the movable pulley system shown,the displacement of the $4 \, kg$ block is twice the displacement of the $8 \, kg$ block,thus $a_1 = 2 a_2$.

If the pulleys shown in the diagram are smooth and massless,and $a_1$ and $a_2$ are the accelerations of the blocks of mass $4 \, kg$ and $8 \, kg$ respectively,then:

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