Determine the relationship that governs the velocities of four cylinders if $v_{A}$ , $v_{B}$ , $v_{C}$ and $v_{D}$ represents velocities of block A, B, C and D. Consider downward velocity as positive and strings inextensible.

4 v_{A} + 8 v_{B} + 4 v_{C} + v_{D} = 0

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- 4 v_{A} + 8 v_{B} - 4 v_{C} + v_{D} = 0

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4 v_{A} - 8 v_{B} + 4 v_{C} - v_{D} = 0

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4 v_{A} + 8 v_{B} - 4 v_{C} + v_{D} = 0

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Solution

The correct option is A $4 v_{A} + 8 v_{B} + 4 v_{C} + v_{D} = 0$
Let the velocities of pulleys $P_{1}$ , $P_{2}$ , $P_{3}$ , and $P_{4}$ be $v_{P_{1}}$ , $v_{P_{2}}$ , $v_{P_{3}}$ and $v_{P_{4}}$ respectively.
$v_{p_{2}} = v_{C}$ and $v_{p_{4}} = v_{B}$ . All velocities are assumed to be in downward direction.
(a) $l_{1} + l_{2} + l_{3} = constant$
Differentiating w.r.t time, we get
$\Rightarrow v_{D} + v_{p_{1}} + v_{p_{1}} = 0$
$\Rightarrow 2 v_{p_{1}} = - v_{D}$ ... (i)
(b) $l_{4} + l_{5} + l_{6} + l_{7} = constant$
Differentiating w.r.t time, we get
$- v_{p_{1}} + v_{C} + v_{C} + v_{p_{3}} + v_{p_{3}} = 0$
$\Rightarrow 2 v_{p_{3}} + 2 v_{C} = v_{p_{1}}$ ... (ii)
(c) $l_{8} + l_{9} + l_{10} = constant$
Differentiating w.r.t time, we get
$- v_{p_{3}} + v_{B} + v_{B} + v_{A} = 0$
$\Rightarrow 2 v_{B} + v_{A} = v_{p_{3}}$ ... (iii)
From (i), (ii) and (iii)
$4 v_{B} + 2 v_{A} + 2 v_{C} = - \frac{v_{D}}{2}$
$\Rightarrow 4 v_{A} + 8 v_{B} + 4 v_{C} + v_{D} = 0$

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Determine the relationship that governs the velocities of four cylinders if vA, vB, vC and vD represents velocities of block A, B, C and D. Consider downward velocity as positive and strings inextensible.

Determine the relationship that governs the velocities of four cylinders if $v_{A}$ , $v_{B}$ , $v_{C}$ and $v_{D}$ represents velocities of block A, B, C and D. Consider downward velocity as positive and strings inextensible.